4-1
Chapter 4
Further issues with the
classical linear regression model
4-2
本章目标
继续讨论古典线性回归模型
? 了解确定模型优劣的各种方法
? 普通最小二乘法 OLS可能遇到的各种问题及其处理
4-3
1 Goodness of Fit Statistics
? We would like some measure of how well our regression model
actually fits the data,*
? We have goodness of fit statistics to test this,i.e,how well the
sample regression function (srf) fits the data.
? The most common goodness of fit statistic is known as R2,One
way to define R2 is to say that it is the square of the correlation
coefficient between y and,
? For another explanation,recall that what we are interested in
doing is explaining the variability of y about its mean value,,
i.e,the total sum of squares,TSS总变差,
? We can split the TSS into two parts,the part which we have
explained (known as the explained sum of squares,ESS) and
the part which we did not explain using the model (the RSS)*.
$y
? ?? ??
t t
yyT S S 2
y
4-4
Defining R2
? That is,TSS = ESS + RSS
? Goodness of fit statistic is
? R2 must always lie between zero and one,To understand
this,consider two extremes
RSS = TSS i.e,ESS = 0 so R2 = ESS / TSS = 0
ESS = TSS i.e,RSS = 0 so R2 = ESS / TSS = 1
R E S STS S2 ?
R E S ST S S T S S R S ST S S R S ST S S2 1? ? ? ? ?
? ? ? ? ?? ? ????
t
t
t t
tt uyyyy
222 ??
4-5
The Limit Cases,R2 = 0 and R2 = 1
t
y
y
t
x
t
y
t
x
4-6
Problems with R2 as a Goodness of Fit
Measure
1,R2 is defined in terms of variation about the mean of y so
that if a model is reparameterised (rearranged) and the
dependent variable changes,R2 will change.
2,R2 never falls if more regressors are added to the
regression,e.g,consider:
Regression 1,yt = ?1 + ?2x2t + ?3x3t + ut
Regression 2,y = ?1 + ?2x2t + ?3x3t + ?4x4t + ut
R2 will always be at least as high for regression 2 relative to
regression 1.
3,R2 quite often takes on values of 0.9 or higher for time
series regressions.
4-7
Adjusted R2
? In order to get around these problems,a modification is often
made which takes into account the loss of degrees of freedom
associated with adding extra variables,This is known as,or
adjusted R2:
? So if we add an extra regressor,k increases and unless R2
increases by a more than offsetting amount,will actually fall,
? 可用于决定某一变量是否应包括在模型中。
? There are still problems with the criterion:
1,A,soft” rule。如果只按这一标准选择模型，模型中会
包含很多边际显著或不显著的变量。
2,No distribution for or R2。从而不能进行假设检验，
以比较一个模型的拟合优度是否显著高于另一个模型。
2R
?????? ??
??? )1(11 22 R
kT
TR
2R
2R
2R
4-82 A Regression Example,
Hedonic House Pricing Models
? Hedonic models are used to value real assets,especially
housing,and view the asset as representing a bundle of
characteristics.
? Des Rosiers and Thérialt (1996) consider the effect of various
amenities on rental values for buildings and apartments 5
sub-markets in the Quebec area of Canada.
? The rental value in Canadian Dollars per month (the
dependent variable) is a function of 9 to 14 variables
(depending on the area under consideration),The paper
employs 1990 data,and for the Quebec City region,there are
13,378 observations,and the 12 explanatory variables are:
4-9
LnAGE - log of the apparent age of the property
NBROOMS - number of bedrooms
AREABYRM - area per room (in square metres)
ELEVATOR - a dummy variable = 1 if the building has
an elevator; 0 otherwise
BASEMENT - a dummy variable = 1 if the unit is located
in a basement; 0 otherwise
OUTPARK - number of outdoor parking spaces
INDPARK - number of indoor parking spaces
NOLEASE - a dummy variable = 1 if the unit has no
lease租借权 attached to it; 0 otherwise
LnDISTCBD - log of the distance in kilometres to the
central business district
SINGLPAR - percentage of single parent families in the
area where the building stands
DSHOPCNTR - distance in kilometres to the nearest
shopping centre
VACDIFF1 - vacancy difference between the building
and the census figure
4-10
Hedonic House Pricing Models,
? Examine the signs and sizes of the coefficients.
– The coefficient estimates themselves show the
Canadian dollar rental price per month of each
feature of the dwelling,
? The coefficient on the constant term often has little
useful interpretation.
4-11
Hedonic House Price Results
Dependent Variable,Canadian Dollars per Month
Va r iable C oe f f icient t - r a ti o A pri ori si g n e x pe c ted
I nte r c e pt 282.21 56.09 +
L nA GE - 53.10 - 59.71 -
NB R OOMS 48.47 104.81 +
AREAB YRM 3.97 29.99 +
E L EV ATOR 88.51 45.04 +
B AS EMENT - 15.90 - 11.32 -
OUTP ARK 7.17 7.07 +
I N DP ARK 73.76 31.25 +
NO L EAS E - 16,99 - 7.62 -
L nD I S TC B D 5.84 4.60 -
S I NG L P AR - 4.27 - 38.88 -
DS HOP C NTR - 10.04 - 5.97 -
VACD I F F 1 0.29 5.98 -
No tes,A d j u s ted R
2
= 0, 6 5 l; r eg r ess io n F - s tat is tic = 2 0 8 2, 2 7, So u r ce, De s Ro s ier s a n d
T h é r ialt
( 1 9 9 6 ), Rep r in ted w ith p er m is s io n o f t h e Am er ica n Rea l E s t at e So cie t y,
4-12
3 Tests of Non-nested Hypotheses
? All of the hypothesis tests concluded thus far have been in the
context of,nested” models.
? But what if we wanted to compare between the following models?
? We could use R2 or adjusted R2,but what if the number of
explanatory variables were different across the 2 models?
? 还有一些其他的“信息准则”。
? 但很多情况下，不同的准则可能导致选取不同的模型。
? An alternative approach is an encompassing test,based on
examination of the hybrid model:
ttt
ttt
vxy
uxy
???
???
321
221
:2M ode l
:1M ode l
??
??
tttt wxxy ???? 33221:3M o d e l ???
4-13
Tests of Non-nested Hypotheses
? There are 4 possible outcomes when Model 3 is estimated:
– ?2 is significant but ?3 is not,model 1
– ?3 is significant but ?2 is not,model 2
– ?2 and ?3 are both statistically significant,Model 3
– Neither ?2 nor ?3 are significant,none; other method
? Problems with encompassing approach
– Hybrid model may be meaningless
– Possible high correlation between x2 and x3.
4-14
4 Violation of the Assumptions of the
CLRM
? Recall that we assumed of the CLRM disturbance terms:
1,E(ut) = 0
2,Var(ut) = ?2 < ?
3,Cov (ui,uj) = 0
4,The X matrix is non-stochastic or fixed in repeated
samples
5,ut ? N(0,?2)
4-15
Investigating Violations of the
Assumptions of the CLRM
? We will now study these assumptions further,
- How we test for violations
- Causes
- Consequences in general we could encounter any
combination of 3 problems:
- the coefficient estimates are wrong
- the associated standard errors are wrong
- the distribution that we assumed for the
test statistics will be inappropriate
- Solutions
- the assumptions are no longer violated
- we work around the problem so that we
use alternative techniques which are still valid
4-16Statistical Distributions for Diagnostic
Tests
? Often,an F- and a ?2- version of the test are available.
? The F-test version(Wald test) involves estimating a restricted
and an unrestricted version of a test regression and
comparing the RSS.
? The ?2- version is sometimes called an,LM” test,and only
has one degree of freedom parameter,the number of
restrictions being tested,m,
? Asymptotically,the 2 tests are equivalent since the ?2 is a
special case of the F-distribution:
? For small samples,the F-version is preferable.
? ? ? ? ????? kTkTmF
m
m as,2?
4-17
5 Assumption 1,E(ut) = 0
? Assumption that the mean of the disturbances is zero.
? For all diagnostic tests,we cannot observe the
disturbances and so perform the tests of the residuals.
? The mean of the residuals will always be zero provided
that there is a constant term in the regression.
? 没有常数项时，可能导致斜率估计值出现偏差。
? R2 可能是负的。
4-18
6 Assumption 2,Var(ut) = ?2 < ?
? We have so far assumed that the variance of the errors is
constant,?2 - this is known as homoscedasticity,If the
errors do not have a constant variance,we say that they
are heteroscedastic e.g,say we estimate a regression and
calculate the residuals,.$ut
t
u? +
-
t
x
2
4-19
Detection of Heteroscedasticity
? Graphical methods
? Formal tests:
One of the best is White’s general test for heteroscedasticity.
? The test is carried out as follows:
1,Assume that the regression we carried out is as follows
yt = ?1 + ?2x2t + ?3x3t + ut
And we want to test Var(ut) = ?2,We estimate the model,
obtaining the residuals,
2,Then run the auxiliary regression
$ut
tttttttt vxxxxxxu ??????? 326235224332212? ??????
4-20Performing White’s Test for
Heteroscedasticity
3,Obtain R2 from the auxiliary regression and multiply it by
the number of observations,T,It can be shown that
T R2 ??2 (m)
where m is the number of regressors in the auxiliary
regression excluding the constant term.
(还可以对辅助回归方程做 F 检验 )
4,If the ?2 test statistic from step 3 is greater than the
corresponding value from the statistical table then reject the
null hypothesis that the disturbances are homoscedastic.
? 例子,p150
4-21
Consequences of Using OLS in the
Presence of Heteroscedasticity
? OLS estimation still gives unbiased coefficient estimates,but
they are no longer BLUE.
? This implies that if we still use OLS in the presence of
heteroscedasticity,our standard errors could be inappropriate
and hence any inferences we make could be misleading.
? Whether the standard errors calculated using the usual
formulae are too big or too small will depend upon the form of
the heteroscedasticity.
4-22How Do we Deal with
Heteroscedasticity?
? If the form (i.e,the cause) of the heteroscedasticity is known,
then we can use an estimation method which takes this into
account (called generalised least squares,GLS).
? A simple illustration of GLS is as follows,Suppose that the
error variance is related to another variable zt by
? To remove the heteroscedasticity,divide the regression
equation by zt
Now
for known zt.
? ? 22v ar tt zu ??
t
t
t
t
t
tt
t v
z
x
z
x
zz
y ???? 3
3
2
21
1 ???
t
tt
z
uv ?
? ? ? ? 22 222v a rv a rv a r ?? ?????
?
?
???
??
t
t
t
t
t
t
t z
z
z
u
z
uv
4-23
Other Approaches to Dealing
with Heteroscedasticity
? So the disturbances from the new regression equation will be
homoscedastic.
? Other solutions (异方差的形式未知 ) include:
1,Transforming the variables into logs* or reducing by some
other measure of,size”.
2,Use White’s heteroscedasticity consistent相容 standard error
estimates.
The effect of using White’s correction is that in general the
standard errors for the slope coefficients are increased relative
to the usual OLS standard errors.
This makes us more,conservative” in hypothesis testing,so that
we would need more evidence against the null hypothesis before
we would reject it.
4-24
用 EViews检验和处理异方差
? White异方差检验
? p153
? White修正标准差估计
? p154-155
4-25
7 Assumption 3,cov(ui,uj) = 0 for i≠j
Background– The Concept of a Lagged Value
t yt yt-1 ?yt
1989M09 0.8 - -
1989M10 1.3 0.8 1.3-0.8 = 0.5
1989M11 -0.9 1.3 -0.9-1.3 = -2.2
1989M12 0.2 -0.9 0.2- -0.9= 1.1
1990M01 -1.7 0.2 -1.7-0.2 = -1.9
1990M02 2.3 -1.7 2.3- -1.7= 4.0
1990M03 0.1 2.3 0.1-2.3= -2.2
1990M04 0.0 0.1 0.0-0.1= -0.1
.,,,
.,,,
.,,,
4-26
Test for autocorrelation
? We assumed of the CLRM’s errors that Cov (ui,uj) = 0 for i?j,
i.e,This is essentially the same as saying there is no pattern in
the errors.
? Obviously we never have the actual u’s,so we use their sample
counterpart,the residuals (the ’s).
? If there are patterns in the residuals from a model,we say that
they are autocorrelated or serially correlated.
? Some stereotypical patterns we may find in the residuals are
given on the next 3 slides.
$ut
4-27
Positive Autocorrelation
Positive autocorrelation is indicated by a cyclical
residual plot over time.
+
-
-
t
u?
+
1
?
?t
u
- 3, 7
-6
- 6, 5
-6
- 3, 1
-5
-3
0, 5
-1
1
4
3
5
7
8
7
+
-
T i m e
t
u?
4-28
Negative Autocorrelation
Negative autocorrelation is indicated by an alternating pattern
where the residuals cross the time axis more frequently than
if they were distributed randomly
+
-
-
t
u?
+
1
?
?t
u
+
-
t
u?
T i m e
4-29
No pattern in residuals –
No autocorrelation
No pattern in residuals at all,this is what we would like to see
+
t
u?
-
-
+
1
?
?t
u
+
-
t
u?
4-30
Detecting Autocorrelation:
The Durbin-Watson Test
The Durbin-Watson (DW) is a test for first order
autocorrelation - i.e,it assumes that the relationship is between
an error and the previous one
ut = ?ut-1 + vt (1)
where vt ? N(0,?v2).
? The DW test statistic actually tests
H0, ? = 0 and H1, ?? 0
? The test statistic is calculated by
?
?有误
? ?
DW
u u
u
t t
t
T
t
t
T?
? ?
?
?
?
?
$ $
$
1
2
2
2
2
4-31
The Durbin-Watson Test,
Critical Values
? We can also write *
(2)
where is the estimated correlation coefficient,Since is a
correlation,it implies that,
? Rearranging for DW from (2) would give 0?DW?4.
If = 0,DW = 2,So roughly speaking,do not reject the null
hypothesis if DW is near 2 ? i.e,there is little evidence of
autocorrelation
? Unfortunately,DW has 2 critical values,an upper critical value
(du) and a lower critical value (dL),and there is also an
intermediate region where we can neither reject nor not reject
H0.
DW ? ?2 1( $ )?
$?
$?
1?1 ??? p
$?
4-32
The Durbin-Watson Test,Interpreting
the Results
Conditions which Must be Fulfilled for DW to be a Valid Test
1,Constant term in regression
2,Regressors are non-stochastic
3,No lags of dependent variable
4-33Another Test for Autocorrelation,
The Breusch-Godfrey Test
? It is a more general test for rth order autocorrelation:
?N(0,)
? The null and alternative hypotheses are:
H0, ?1 = 0 and ?2 = 0 and,.,and ?r = 0
H1, ?1 ? 0 or ?2 ? 0 or,.,or ?r ? 0
? The test is carried out as follows:
1,Estimate the linear regression using OLS and obtain the
residuals,.
2,Regress on all of the regressors from stage 1 (the x’s) plus
,Obtain R2 from this regression.
3,It can be shown that (T-r)R2 ??2(r)
$ut
$ut
u u u u u v vt t t t r t r t t? ? ? ? ? ?? ? ? ?? ? ? ?1 1 2 2 3 3,,,,
$,$,...,$u u ut t t r? ? ?1 2
2v?
4-34
The Breusch-Godfrey Test
? If the test statistic exceeds the critical value from the statistical
tables,reject the null hypothesis of no autocorrelation.
? The test is more general than the DW test.
? 这种检验的一个潜在困难是如何确定残差适当的滞后值 r。对
此没有明确的、一致的答案。可以对一组滞后值分别进行检
验，还常常考虑到使用数据的频率。例如，对于月度或季度
数据，滞后值 r可以取 12或 4。
4-35Consequences of Ignoring
Autocorrelation if it is Present
? The coefficient estimates derived using OLS are still unbiased,
but they are inefficient,i.e,they are not BLUE,even in large
sample sizes.
? Thus,if the standard error estimates are inappropriate,there
exists the possibility that we could make the wrong inferences.
? R2 is likely to be inflated relative to its,correct” value for
positively correlated residuals.
4-36
,Remedies” for Autocorrelation
? If the form of the autocorrelation is known,we could use a
GLS procedure – i.e,an approach that allows for
autocorrelated residuals e.g.,Cochrane-Orcutt procedure*.
? 在大样本情况下, 利用 D-W统计量, ?≈ 1－ d／ 2
? But such procedures that,correct” for autocorrelation require
assumptions about the form of the autocorrelation.
? If these assumptions are invalid,the cure would be more
dangerous than the disease! - see Hendry and Mizon (1978).
? However,it is unlikely to be the case that the form of the
autocorrelation is known,and a more,modern” view is that
residual autocorrelation presents an opportunity to modify the
regression.
? Newey-West 异方差相容自相关协方差估计
4-37
Dynamic Models
? All of the models we have considered so far have been static,
e.g.
yt = ?1 + ?2x2t +,.,+ ?kxkt + ut
? But we can easily extend this analysis to the case where the
current value of yt depends on previous values of y or one of
the x’s,e.g.(ADL)
yt = ?1 + ?2x2t +,.,+ ?kxkt + ?1yt-1 +?2x2t-1 + … + ?kxkt-1+ ut
? We could extend the model even further by adding extra lags,
e.g,x2t-2,yt-3,
? 加入多少滞后项是一个棘手的问题,相关理论可能有所帮助,
4-38
Models in First Difference Form
? Another way to sometimes deal with the problem of
autocorrelation is to switch to a model in first differences,
? Denote the first difference of yt,i.e,yt - yt-1 as ?yt; similarly for
the x-variables,?x2t = x2t - x2t-1 etc.
? The model would now be
?yt = ?1 + ?2 ?x2t +,.,+ ?k?xkt + ut
? Sometimes the change in y is purported to depend on previous
values of y or xt as well as changes in x:
?yt = ?1 + ?2 ?x2t + ?3x2t-1 +?4yt-1 + ut
4-39
产生自相关的原因
? 经济变量的惯性作用
– 消费刚性 ;销售量
? 经济行为的滞后性
– 基础设施建设周期 ;政策的滞后性质
? 随机因素的干扰
– 自然灾害、战争、政策的持续影响
? 设定偏误：所建模型不正确
– 模型函数形式不正确
– 遗漏主要变量
4-40Why Might we Want/Need To Include
Lags in a Regression?
? Inertia of the dependent variable
? Over-reactions(主要针对金融市场 )
? Other problems with the regression could cause the null
hypothesis of no autocorrelation to be rejected:
– Omission of relevant variables,which are themselves
autocorrelated.
– If we have committed a,misspecification” error by
using an inappropriate functional form.
– Autocorrelation resulting from unparameterised
seasonality.
4-41
The Long Run Static Equilibrium Solution
? One interesting property of a dynamic model is its long run or
static equilibrium solution.
?,Equilibrium” implies that the variables have reached some
steady state and are no longer changing,i.e,if y and x are in
equilibrium,we can say
yt = yt+1 =,.,=y and xt = xt+1 =,.,=x
Consequently,?yt = yt - yt-1 = y - y = 0 etc.
? So the way to obtain a long run static solution is:
1,Remove all time subscripts from variables
2,Set error terms equal to their expected values,E(ut)=0
3,Remove first difference terms altogether
4,Gather terms in x together and gather terms in y together.
4-42
The Long Run Static Equilibrium
Solution,An Example
? If our model is
?yt = ?1 + ?2 ?x2t + ?3 x2t-1 +?4 yt-1 + ut
then the static solution would be given by
0 = ?1 + ?3 x2 +?4 y
?4 y = - ?1 - ?3 x2
2
4
3
4
1 xy
?
?
?
? ???
4-43Problems with Adding Lagged
Regressors to,Cure” Autocorrelation
? Inclusion of lagged values of the dependent variable violates
the assumption that the RHS variables are non-stochastic.
在小样本情况下, 将因变量的滞后变量作为解释变量可能导
致参数估计是有偏的 （ 但仍然是一致的 ） 。
? What does an equation with a large number of lags actually
mean?
– 包含滞后或差分项的模型可能难以解释并且无法用于检验
原先的理论 。
? Note that if there is still autocorrelation in the residuals of a
model including lags,then the OLS estimators will not even be
consistent.
4-44
8 假设 4,Xt 是非随机变量
? 可以证明：当存在随机的回归自变量时，只要自变量
与扰动项是不相关的,OLS估计量仍然是无偏的和一
致的。
– 证明见第 178页。
? 如果自变量 与扰动项相关，其处理方法将在第六章
中讨论。
4-45
9 Assumption 5：扰动项服从正态分布
? Why did we need to assume normality for hypothesis testing?
Testing for Departures from Normality
? The Bera 贝拉 Jarque 雅克 normality test
? Skewness and kurtosis are the (standardised) third and fourth
moments of a distribution.
? A normal distribution is not skewed and is defined to have a
coefficient of kurtosis of 3.
? The kurtosis of the normal distribution is 3 so its excess
kurtosis is zero.
4-46
Normal versus Skewed Distributions
A normal distribution A skewed distribution
f( x )
x
x
f( x )
4-47Leptokurtic尖峰 versus Normal
Distribution
-5, 4 -3, 6 -1, 8 -0, 0 1, 8 3, 6 5, 4
0, 0
0, 1
0, 2
0, 3
0, 4
0, 5
4-48
Testing for Normality
? Bera and Jarque formalise this by testing the residuals for
normality by testing whether the coefficient of skewness and the
coefficient of excess kurtosis are jointly zero.
? It can be proved that the coefficients of skewness and kurtosis
can be expressed respectively as:
and
? The Bera Jarque test statistic is given by
? We estimate b1 and b2 using the residuals from the OLS
regression,.*$u
? ?b
E u
1
3
2 3 2
? [ ]/
? ? ?
b E u2
4
2 2
? [ ]
?
? ? ? ?2~
24
3
6
2
2
2
2
1 ??
?
?
?
?
? ??? bbTW
4-49What do we do if we find evidence of
Non-Normality?
? It is not obvious what we should do!
? Could use a method which does not assume normality,but
difficult and what are its properties?
? 当样本容量充分大时, 扰动项的正态性假定不太重要 。
? Often the case that one or two very extreme residuals
causes us to reject the normality assumption.
? An alternative is to use dummy variables.
e.g,say we estimate a monthly model of asset returns from
1980-1990,and we plot the residuals,and find a
particularly large outlier for October 1987:
4-50What do we do if we find evidence
of Non-Normality?
? Create a new variable:
D87M10t = 1 during October 1987 and zero otherwise.
This effectively knocks out that observation,But we need a
theoretical reason for adding dummy variables,*
O c t
1987
+
-
T i m e
t
u?
4-51
10 Multicollinearity
? This problem occurs when the explanatory variables are very
highly correlated with each other.
? Perfect multicollinearity
- e.g,suppose x3 = 2x2,Cannot estimate all the coefficients
? Near Multicollinearity
? Problems if Near Multicollinearity is Present but Ignored
- R2 will be high but the individual coefficients will have high
standard errors.
- The regression becomes very sensitive to small changes in
the specification.
- Thus confidence intervals for the parameters will be very
wide,and significance tests might therefore give
inappropriate conclusions.
4-52
产生多重共线性的可能原因
? 许多经济变量往往存在共同的变化趋势。
– 如经济增长、收入增长、产品销售增长、储蓄增长
? 用截面数据建立回归模型，解释变量常常从经济意义上存在
着密切的关联度。
– 如资本、劳动力、能源、科技进步等都与企业的生产规模密切相关。
? 在模型中较多采用滞后变量
– 滞后变量从经济性质上看与原来的变量并没有区别。
? 解释变量选择不当
– 如化肥使用量、农业生产资金投入同时作为解释变量
4-53
Measuring Multicollinearity
? The easiest way to measure the extent of multicollinearity is
simply to look at the matrix of correlations between the
individual variables,e.g.
? 变量显著性 （ t） 与方程显著性 (R2,F)的综合判断
? But another problem,if 3 or more variables are linear
e.g,x2t + x3t = x4t,it would be very difficult to detect.
? Note that high correlation between y and one of the x’s is not
muticollinearity.
C orr x 2 x 3 x 4
x 2 - 0,2 0,8
x 3 0,2 - 0,3
x 4 0,8 0,3 -
4-54
Solutions to the Problem of
Multicollinearity
? 多重共线性不影响 OLS估计的 BLUE性质 。
?, Traditional” approaches,such as ridge regression 岭回归 or
principal components,But these usually bring more problems
than they solve.
? Some econometricians argue that if the model is otherwise OK,
just ignore it。 特别是用于预测时 。
? The easiest ways to,cure” the problems are
- drop one of the collinear variables。 但须考虑理论的要求 。
- transform the highly correlated variables into a ratio*
- go out and collect more data 增加样本容量 。 e.g.
- a longer run of data
- switch to a higher frequency
Use pooled sample*
? 逐步回归法 *
4-55
11 Adopting the Wrong Functional Form
? We have previously assumed that the appropriate functional
form is linear,This may not always be true.
? We can formally test this using Ramsey’s RESET test
（ regression specification error test ）,which is a
general test for mis-specification of functional form.
? Essentially the method works by adding higher order terms of
the fitted values (e.g,etc.) into an auxiliary regression:
Regress on powers of the fitted values:
The test statistic is given by TR2 and is distributed as
a,
? So if the value of the test statistic is greater than a then
reject the null hypothesis that the functional form was correct.
$ $ $,,, $u y y y vt t t p t p t? ? ? ? ? ??? ? ? ?0 1 2 2 3 1
$,$y yt t2 3
?2 1( )p ?
$ut
?2 1( )p ?
4-56
What do we do if this is the case?
? The RESET test gives us no guide as to what a better
specification might be.
? 对于参数非线性模型, 一般需要使用非线性估计方法 。
? 对于一些参数线性而解释变量非线性的模型, 仍可以使用
OLS进行估计 。
? 例如 if the true model is
In this case the remedy is obvious.
? Another possibility is to transform the data into logarithms.
This will linearise many previously multiplicative models into
additive ones:
ttttt uxxxy ????? 324223221 ????
tttutt uxyeAxy t ????? lnln ???
4-5712 Omission of an Important Variable
13 Inclusion of an Irrelevant Variable
Omission of an Important Variable
? Consequence,The estimated coefficients on all the other
variables will be biased and inconsistent unless the
excluded variable is uncorrelated with all the included
variables.
? Even if this condition is satisfied,the estimate of the
coefficient on the constant term will be biased.
? The standard errors will also be biased.
Inclusion of an Irrelevant Variable
? Coefficient estimates will still be consistent and unbiased,
but the estimators will be inefficient.
4-58
14 Parameter Stability Tests
? So far,we have estimated regressions such as
? We have implicitly assumed that the parameters (?1,?2 and ?3)
are constant for the entire sample period.
? We can test this implicit assumption using parameter stability
tests,The idea is essentially to split the data into sub-periods
and then to estimate up to three models,for each of the sub-
parts and for all the data and then to,compare” the RSS of the
models.
? There are two types of test we can look at:
- Chow test (analysis of variance test)
- Predictive failure tests
y t = ? 1 + ? 2 x 2 t + ? 3 x 3 t + u t
4-59
The Chow邹至庄 Test
? The steps involved are:
1,Split the data into two sub-periods,Estimate the
regression over the whole period and then for the two sub-
periods separately (3 regressions),Obtain the RSS for each
regression.
2,The restricted regression is now the regression for the
whole period while the,unrestricted regression” comes in
two parts,for each of the sub-samples.
We can thus form an F-test which is the difference between
the RSS’s,The statistic is
? ?R S S R S S R S S
R S S R S S
T k
k
? ?
? ?
?1 2
1 2
2
4-60
The Chow邹至庄 Test
RSS = RSS for whole sample
RSS1 = RSS for sub-sample 1
RSS2 = RSS for sub-sample 2
T = number of observations
2k = number of regressors in the,unrestricted” regression
(since it comes in two parts)
k = number of regressors in (each part of the),unrestricted”
regression
? 3,Perform the test,If the value of the test statistic is greater
than the critical value from the F-distribution,which is an
F(k,T-2k),then reject the null hypothesis that the
parameters are stable over time.*
4-61
A Chow Test Example
? Consider the following regression for the CAPM ? (again) for
the returns on Glaxo.
? Say that we are interested in estimating Beta for monthly data
from 1981-1992,The model for each sub-period is
? 1981M1 - 1987M10
Rgt =0.24 + 1.2RMt T = 82 RSS1 = 0.03555
? 1987M11 - 1992M12
Rgt = 0.68 + 1.53RMt T = 62 RSS2 = 0.00336
? 1981M1 - 1992M12
Rgt = 0.39 + 1.37RMt T = 144 RSS = 0.0434
4-62
A Chow Test Example - Results
? The null hypothesis is
?
= 7.698
? Compare with 5% F(2,140) = 3.06
We reject H0 at the 5% level and say that we reject the
restriction that the coefficients are the same in the two
periods.
H and0 1 2 1 2,? ? ? ?? ?
? ?T e s t s t a t i s t i c ? ? ?
? ?
?0 0434 0 0355 0 00336
0 0355 0 00336
144 4
2
.,,
.,
4-63
The Predictive Failure Test
? Problem with the Chow test is that we need to have enough data
to do the regression on both sub-samples,i.e,T1>>k,T2>>k.
? An alternative formulation is the predictive failure test.
? What we do with the predictive failure test is estimate the
regression over a,long” sub-period (i.e,most of the data) and
then we predict values for the other period and compare the two.
- Run the regression for the whole period (the restricted
regression) and obtain the RSS
- Run the regression for the,large” sub-period and obtain the
RSS (called RSS1),Note we call the number of observations T1
where T2 = number of observations we are attempting to
“predict”,The test statistic will follow an F(T2,T1-k).*
2
1
1
1S t at i s t i cT es t
T
kT
R S S
R S SR S S ????
4-64
Backwards versus Forwards
Predictive Failure Tests
? There are 2 types of predictive failure tests:
- Forward predictive failure tests,where we keep the last few
observations back for forecast testing,e.g,we have
observations for 1970Q1-1994Q4,So estimate the model over
1970Q1-1993Q4 and forecast 1994Q1-1994Q4.
- Backward predictive failure tests,where we attempt to
“back-cast” the first few observations,e.g,if we have data for
1970Q1-1994Q4,and we estimate the model over 1971Q1-
1994Q4 and backcast 1970Q1-1970Q4.
4-65
Predictive Failure Tests – An Example
? We have the following models estimated:
For the CAPM ? on Glaxo(!).
? 1981M1-1992M12
Rgt = 0.39 + 1.37RMt T = 144 RSS = 0.0434
? 1981M1-1990M12
Rgt = 0.32 + 1.31RMt T1 = 120 RSS1 = 0.0420
Can this regression adequately,forecast” the values?
= 0.164
Compare with F(24,118) = 1.66.
So we do not reject the null hypothesis that the model can
adequately predict the observations for the last two years,
24
21 2 0
0 4 2 0.0
0 4 2 0.00 4 3 4.0S t a t i s t i cT e s t ????
4-66
How do we decide the sub-parts to use?
0
200
400
600
800
1000
1200
1400
1
27 53 79
1
0
5
1
3
1
1
5
7
1
8
3
2
0
9
2
3
5
2
6
1
2
8
7
3
1
3
3
3
9
3
6
5
3
9
1
4
1
7
4
4
3
S a m p l e P e ri o d
V
a
l
u
e
o
f
S
e
r
i
e
s
(
y
t
)
? As a rule of thumb,we could use all or some of the following:
- Plot the dependent variable over time and split the data
accordingly to any obvious structural changes in the series,e.g.
4-67
How do we decide the sub-parts to use?
- Split the data according to any known important historical
events (e.g,stock market crash,new government elected)
- Use all but the last few observations and do a forward
predictive failure test on those.
- Use all but the first few observations and do a backward
predictive failure test on those.
? 如果参数 稳定性检验被拒绝, 则可采用
– 重新设定模型, 如引入其它变量
– 对子样本分别进行参数估计
? Eviews中的稳定性检验（参见 p206-207）
4-68
15 A Strategy for Building
Econometric Models
Our Objective:
? To build a statistically adequate empirical model which
- satisfies the assumptions of the CLRM
- is parsimonious
- has the appropriate theoretical interpretation
- has the right,shape” - i.e.
- all signs on coefficients are,correct”
- all sizes of coefficients are,correct”
- is capable of explaining the results of all competing
models
4-6915 Approaches to Building
Econometric Models
? There are 2 popular philosophies of building econometric models,the
“specific-to-general” and,general-to-specific” approaches.
?,Specific-to-general” was used almost universally until the mid 1980’s,
and involved starting with the simplest model and gradually adding to it.
? Little,if any,diagnostic testing was undertaken,But this meant that all
inferences were potentially invalid.
? 如果初始模型被错误设定, 诊断检验也不一定能可靠指出问题的来源 。
? An alternative and more modern approach to model building is the
“LSE” or Hendry,general-to-specific” methodology.
? The advantages of this approach are that it is statistically sensible and
also the theory on which the models are based usually has nothing to say
about the lag structure of a model.因此,模型的滞后结构主要由数据本
身来确定 。 此外, 更注重考虑遗漏相关变量的后果 。
? 这种方法认为, 一个好的模型与数据和理论都是一致的；而且应包含
竞争 rival模型 。 建议广泛使用诊断检验以保证模型的统计准确性 。
4-70
Hendry提出的模型标准
最终的可接受模型应当满足以下一些标准：
? 逻辑上合理
? 与理论一致，包括满足相关的参数约束
? 回归自变量与误差项不相关
? 参数估计值在整个样本区间上是稳定的
? 残差是白噪声（完全随机的）
? 能够解释所有竞争模型的结果，并给出更多的解释 (包容原则 )
– 简约包容 parsimonious encompassing：小模型能解释大模型的所有
结果
4-71
The General-to-Specific Approach
? First step is to form a,large” model with lots of variables on the
right hand side。 GUM (generalised unrestricted model)
? At this stage,we want to make sure that the model satisfies all of
the assumptions of the CLRM
? If the assumptions are violated,we need to take appropriate
actions to remedy this,e.g.
- taking logs
- adding lags
- dummy variables
? We need to do this before hypotheses test
? Once we have a model which satisfies the assumptions,it could
be very big with lots of lags & independent variables
4-72
The General-to-Specific Approach:
Reparameterising the Model
? The next stage is to reparameterise the model by
- knocking out very insignificant regressors
- some coefficients may be insignificantly different from
each other,so we can combine them.
? At each stage,we need to check the assumptions are still OK.
? Hopefully at this stage,we have a statistically adequate
empirical model which we can use for
- testing underlying financial theories
- forecasting future values of the dependent variable
- formulating policies,etc.
4-73
16 Regression Analysis Example:
Determinants of Sovereign Credit Ratings
? Cantor and Packer (1996)
Financial background:
? What are sovereign credit ratings and why are we interested in them?
? Two ratings agencies (Moody’s and Standard and Poor’s) provide credit
ratings for many governments.
? Each possible rating is denoted by a grading:
Moody’s Standard and Poor’s
Aaa AAA
…… …,.
B3 B-
4-74
Purposes of the Paper
- to attempt to explain and model how the ratings agencies arrived at
their ratings.
- to use the same factors to explain the spreads of sovereign yields
above a risk-free proxy
- to determine what factors affect how the sovereign yields react to
ratings announcements
4-75
Determinants of Sovereign Ratings
? Data
Quantifying the ratings (dependent variable),Aaa/AAA=16,...,B3/B-=1
? Explanatory variables (units of measurement):
- Per capita income in 1994 (thousands of dollars)
- Average annual GDP growth 1991-1994 (%)
- Average annual inflation 1992-1994 (%)
- Fiscal balance,Average annual government budget surplus as a
proportion of GDP 1992-1994 (%)
- External balance,Average annual current account surplus as a proportion
of GDP 1992-1994 (%)
- External debt Foreign currency debt as a proportion of exports 1994 (%)
- Dummy for economic development
- Dummy for default history
Income and inflation are transformed to their logarithms.
4-76
The model,Linear and estimated using OLS
Dep en d en t V ar i ab le
Expl an at o r y V a r i a bl e
Expec t ed
si g n
Av er ag e
R at i ng
M ood y ’ s
R at i ng
S& P
R at i ng
M ood y ’ s / S& P
Dif f er e nce
I nt er cep t? 1.442
( 0.663 )
3.408
( 1.379 )
- 0.524
( - 0.223 )
3.932* *
( 2.521 )
Per c api t a i nco m e + 1.242** *
( 5.302 )
1.027** *
( 4.041 )
1.458 ***
( 6.048 )
- 0.431 ***
( - 2.688 )
GD P g r owt h + 0.151
( 1.935 )
0.130
( 1.545 )
0.171 **
( 2.132 )
- 0.040
( 0.756 )
I nf l at i on - - 0.611 ***
( - 2.839 )
- 0.630 ***
( - 2.701 )
- 0.591 ***
( 2.671 )
- 0.039
( - 0.265 )
Fi sc a l B a l an ce + 0.073
( 1.324 )
0.049
( 0.818 )
0.097 *
( 1.71 )
- 0.048
( - 1.274 )
Ext er nal B al anc e + 0.003
( 0.314 )
0.006
( 0.535 )
0.001
( 0.046 )
0.006
( 0.779 )
Ext er nal D ebt - - 0.013 ***
( - 5.088 )
- 0.015 ***
( - 5.365 )
- 0.011 ***
( - 4.236 )
- 0.004 ***
( - 2.133 )
Dev el opm ent dum m y + 2.7 76 * **
( 4.25 )
2,957* * *
( 4.175 )
2.595 ***
( 3.861 )
0.362
( 0.81 )
Def au l t d um m y - - 2.042 ***
( - 3.175 )
- 1.63* *
( - 2.097 )
- 2.622 ***
( - 3.962 )
1.159** *
( 2.632 )
Adj us t ed R
2
0.924 0.905 0.926 0.836
No tes,t - r atio s i n p ar en t h e s es ; *,* *,an d * * * in d ica te s i g n i f ica n ce at t h e 1 0 %,5 % a n d 1 % lev el s
r esp ec tiv el y, So u r ce, Ca n to r an d P ac k er ( 1 9 9 6 ), Re p r in ted w it h p er m i s s io n f r o m I n s t itu ti o n a l I n ve s to r,
4-77
Interpreting the Model
From a statistical perspective
? Virtually no diagnostics
? Adjusted R2 is high
? Look at the residuals,actual rating - fitted rating
From a financial perspective
? Do the coefficients have their expected signs and sizes?
Do Ratings Add to Publicly Available Information?
? Now dependent variable is
- Log (Yield on the sovereign bond - yield on a US treasury bond)
4-78
Do Ratings Add to Publicly Available Available
Information? Results
De pe nde nt V a r iable,L o g ( y ield s p r e a d )
Va r iable Ex pe c ted S ig n ( 1) ( 2) ( 3)
I nte r c e pt? 2.105***
( 16.148)
0.466
( 0.345)
0.074
( 0.071)
Ave r a ge
R a ti ng
- - 0.221***
( - 19.175)
- 0.218***
( - 4.276)
Per c api t a
i nco m e
- - 0.144
( - 0.927)
0.226
( 1.523)
GD P g r owt h - - 0.004
( - 0.142)
0.029
( 1.227)
I nf l at i on + 0.108
( 1.393)
- 0.004
( - 0.068)
Fi sc a l B a l an ce - - 0.037
( - 1.557)
- 0.02
( - 1.045)
Ext er nal
B al anc e
- - 0.038
( - 1.29)
- 0.023
( - 1.008)
Ext er nal D ebt + 0.003***
( 2.651)
0.000
( 0.095)
Dev el opm ent
dumm y
- - 0.723***
( - 2.059)
- 0.38
( - 1.341)
Def au l t d um m y + 0.612***
( 2.577)
0.085
( 0.385)
Adj us t ed R
2
0.919 0.857 0.914
No tes,t - r atio s i n p ar en t h e s es ; *,* *,an d * * * in d ica te s i g n i f ica n ce at t h e 1 0 %,5 % a n d 1 % lev el s
r esp ec tiv el y, So u r ce, Ca n to r an d P ac k er ( 1 9 9 6 ), Re p r in ted w it h p er m i s s io n f r o m I n s t itu ti o n a l I n ve s to r,
4-79
What Determines How the Market Reacts
to Ratings Announcements?
? The sample,Every announcement of a ratings change that occurred between
1987 and 1994 - 79 such announcements spread over 18 countries.
? 39 were actual ratings changes
? 40 were,watchlist / outlook”changes
? The dependent variable,changes in the relative spreads over the US T-bond
over a 2-day period at the time of the announcement.
4-80
What Determines How the Market Reacts
to Ratings Announcements? Explanatory variables.
0 /1 dummies for
- Whether the announcement was positive
- Whether there was an actual ratings change
- Whether the bond was speculative grade
- Whether there had been another ratings announcement in the previous 60 days.
and
- The change in the spread over the previous 60 days.
- The ratings gap between the announcing and the other agency
4-81
What Determines How the Market Reacts
to Ratings Announcements? Results
D e pe nde nt V a ria ble, L o g Re la ti ve Spre a d
I nd e pe nde nt v a ria ble Coef fic ie nt ( t - ra ti o)
I nte rcept - 0.02
( - 1.4)
P osit ive a nnounc e me nts 0.01
(0,34)
Rating s c ha n ge s - 0.01
( - 0.37)
Mood y ’s a nnounc e m e nts 0.02
(1,51)
S pe c ula ti ve g ra d e 0.03**
(2,33)
Chang e in re la ti ve spr e a d s fr om da y – 60 to da y - 1 - 0.06
( - 1.1)
Rating g a p 0.03*
(1,7)
O the r ra ti n g a nnoun c e me nts fr om da y – 60 to da y - 1 0.05**
(2,15)
A djust e d R
2
0.12
No te,* an d * * d en o te s i g n i f i can ce at t h e 1 0 % an d 5 % le v els r esp ectiv el y, S o u r ce,C an t o r an d P ack e r
( 1 9 9 6 ), R e p r in ted w ith p er m is s io n f r o m I n s titu tio n a l I n vesto r,
4-82
Conclusions
? 6 factors appear to play a big role in determining sovereign credit ratings -
incomes,GDP growth,inflation,external debt,industrialised or not,and default
history.
? The ratings provide more information on yields than all of the macro factors put
together.
? We cannot determine well what factors influence how the markets will react to
ratings announcements.
4-83
Comments on the Paper
? Only 49 observations for first set of regressions and 35 for yield regressions and
up to 10 regressors
? No attempt at reparameterisation
? Little attempt at diagnostic checking
? Where did the factors (explanatory variables) come from?
Chapter 4
Further issues with the
classical linear regression model
4-2
本章目标
继续讨论古典线性回归模型
? 了解确定模型优劣的各种方法
? 普通最小二乘法 OLS可能遇到的各种问题及其处理
4-3
1 Goodness of Fit Statistics
? We would like some measure of how well our regression model
actually fits the data,*
? We have goodness of fit statistics to test this,i.e,how well the
sample regression function (srf) fits the data.
? The most common goodness of fit statistic is known as R2,One
way to define R2 is to say that it is the square of the correlation
coefficient between y and,
? For another explanation,recall that what we are interested in
doing is explaining the variability of y about its mean value,,
i.e,the total sum of squares,TSS总变差,
? We can split the TSS into two parts,the part which we have
explained (known as the explained sum of squares,ESS) and
the part which we did not explain using the model (the RSS)*.
$y
? ?? ??
t t
yyT S S 2
y
4-4
Defining R2
? That is,TSS = ESS + RSS
? Goodness of fit statistic is
? R2 must always lie between zero and one,To understand
this,consider two extremes
RSS = TSS i.e,ESS = 0 so R2 = ESS / TSS = 0
ESS = TSS i.e,RSS = 0 so R2 = ESS / TSS = 1
R E S STS S2 ?
R E S ST S S T S S R S ST S S R S ST S S2 1? ? ? ? ?
? ? ? ? ?? ? ????
t
t
t t
tt uyyyy
222 ??
4-5
The Limit Cases,R2 = 0 and R2 = 1
t
y
y
t
x
t
y
t
x
4-6
Problems with R2 as a Goodness of Fit
Measure
1,R2 is defined in terms of variation about the mean of y so
that if a model is reparameterised (rearranged) and the
dependent variable changes,R2 will change.
2,R2 never falls if more regressors are added to the
regression,e.g,consider:
Regression 1,yt = ?1 + ?2x2t + ?3x3t + ut
Regression 2,y = ?1 + ?2x2t + ?3x3t + ?4x4t + ut
R2 will always be at least as high for regression 2 relative to
regression 1.
3,R2 quite often takes on values of 0.9 or higher for time
series regressions.
4-7
Adjusted R2
? In order to get around these problems,a modification is often
made which takes into account the loss of degrees of freedom
associated with adding extra variables,This is known as,or
adjusted R2:
? So if we add an extra regressor,k increases and unless R2
increases by a more than offsetting amount,will actually fall,
? 可用于决定某一变量是否应包括在模型中。
? There are still problems with the criterion:
1,A,soft” rule。如果只按这一标准选择模型，模型中会
包含很多边际显著或不显著的变量。
2,No distribution for or R2。从而不能进行假设检验，
以比较一个模型的拟合优度是否显著高于另一个模型。
2R
?????? ??
??? )1(11 22 R
kT
TR
2R
2R
2R
4-82 A Regression Example,
Hedonic House Pricing Models
? Hedonic models are used to value real assets,especially
housing,and view the asset as representing a bundle of
characteristics.
? Des Rosiers and Thérialt (1996) consider the effect of various
amenities on rental values for buildings and apartments 5
sub-markets in the Quebec area of Canada.
? The rental value in Canadian Dollars per month (the
dependent variable) is a function of 9 to 14 variables
(depending on the area under consideration),The paper
employs 1990 data,and for the Quebec City region,there are
13,378 observations,and the 12 explanatory variables are:
4-9
LnAGE - log of the apparent age of the property
NBROOMS - number of bedrooms
AREABYRM - area per room (in square metres)
ELEVATOR - a dummy variable = 1 if the building has
an elevator; 0 otherwise
BASEMENT - a dummy variable = 1 if the unit is located
in a basement; 0 otherwise
OUTPARK - number of outdoor parking spaces
INDPARK - number of indoor parking spaces
NOLEASE - a dummy variable = 1 if the unit has no
lease租借权 attached to it; 0 otherwise
LnDISTCBD - log of the distance in kilometres to the
central business district
SINGLPAR - percentage of single parent families in the
area where the building stands
DSHOPCNTR - distance in kilometres to the nearest
shopping centre
VACDIFF1 - vacancy difference between the building
and the census figure
4-10
Hedonic House Pricing Models,
? Examine the signs and sizes of the coefficients.
– The coefficient estimates themselves show the
Canadian dollar rental price per month of each
feature of the dwelling,
? The coefficient on the constant term often has little
useful interpretation.
4-11
Hedonic House Price Results
Dependent Variable,Canadian Dollars per Month
Va r iable C oe f f icient t - r a ti o A pri ori si g n e x pe c ted
I nte r c e pt 282.21 56.09 +
L nA GE - 53.10 - 59.71 -
NB R OOMS 48.47 104.81 +
AREAB YRM 3.97 29.99 +
E L EV ATOR 88.51 45.04 +
B AS EMENT - 15.90 - 11.32 -
OUTP ARK 7.17 7.07 +
I N DP ARK 73.76 31.25 +
NO L EAS E - 16,99 - 7.62 -
L nD I S TC B D 5.84 4.60 -
S I NG L P AR - 4.27 - 38.88 -
DS HOP C NTR - 10.04 - 5.97 -
VACD I F F 1 0.29 5.98 -
No tes,A d j u s ted R
2
= 0, 6 5 l; r eg r ess io n F - s tat is tic = 2 0 8 2, 2 7, So u r ce, De s Ro s ier s a n d
T h é r ialt
( 1 9 9 6 ), Rep r in ted w ith p er m is s io n o f t h e Am er ica n Rea l E s t at e So cie t y,
4-12
3 Tests of Non-nested Hypotheses
? All of the hypothesis tests concluded thus far have been in the
context of,nested” models.
? But what if we wanted to compare between the following models?
? We could use R2 or adjusted R2,but what if the number of
explanatory variables were different across the 2 models?
? 还有一些其他的“信息准则”。
? 但很多情况下，不同的准则可能导致选取不同的模型。
? An alternative approach is an encompassing test,based on
examination of the hybrid model:
ttt
ttt
vxy
uxy
???
???
321
221
:2M ode l
:1M ode l
??
??
tttt wxxy ???? 33221:3M o d e l ???
4-13
Tests of Non-nested Hypotheses
? There are 4 possible outcomes when Model 3 is estimated:
– ?2 is significant but ?3 is not,model 1
– ?3 is significant but ?2 is not,model 2
– ?2 and ?3 are both statistically significant,Model 3
– Neither ?2 nor ?3 are significant,none; other method
? Problems with encompassing approach
– Hybrid model may be meaningless
– Possible high correlation between x2 and x3.
4-14
4 Violation of the Assumptions of the
CLRM
? Recall that we assumed of the CLRM disturbance terms:
1,E(ut) = 0
2,Var(ut) = ?2 < ?
3,Cov (ui,uj) = 0
4,The X matrix is non-stochastic or fixed in repeated
samples
5,ut ? N(0,?2)
4-15
Investigating Violations of the
Assumptions of the CLRM
? We will now study these assumptions further,
- How we test for violations
- Causes
- Consequences in general we could encounter any
combination of 3 problems:
- the coefficient estimates are wrong
- the associated standard errors are wrong
- the distribution that we assumed for the
test statistics will be inappropriate
- Solutions
- the assumptions are no longer violated
- we work around the problem so that we
use alternative techniques which are still valid
4-16Statistical Distributions for Diagnostic
Tests
? Often,an F- and a ?2- version of the test are available.
? The F-test version(Wald test) involves estimating a restricted
and an unrestricted version of a test regression and
comparing the RSS.
? The ?2- version is sometimes called an,LM” test,and only
has one degree of freedom parameter,the number of
restrictions being tested,m,
? Asymptotically,the 2 tests are equivalent since the ?2 is a
special case of the F-distribution:
? For small samples,the F-version is preferable.
? ? ? ? ????? kTkTmF
m
m as,2?
4-17
5 Assumption 1,E(ut) = 0
? Assumption that the mean of the disturbances is zero.
? For all diagnostic tests,we cannot observe the
disturbances and so perform the tests of the residuals.
? The mean of the residuals will always be zero provided
that there is a constant term in the regression.
? 没有常数项时，可能导致斜率估计值出现偏差。
? R2 可能是负的。
4-18
6 Assumption 2,Var(ut) = ?2 < ?
? We have so far assumed that the variance of the errors is
constant,?2 - this is known as homoscedasticity,If the
errors do not have a constant variance,we say that they
are heteroscedastic e.g,say we estimate a regression and
calculate the residuals,.$ut
t
u? +
-
t
x
2
4-19
Detection of Heteroscedasticity
? Graphical methods
? Formal tests:
One of the best is White’s general test for heteroscedasticity.
? The test is carried out as follows:
1,Assume that the regression we carried out is as follows
yt = ?1 + ?2x2t + ?3x3t + ut
And we want to test Var(ut) = ?2,We estimate the model,
obtaining the residuals,
2,Then run the auxiliary regression
$ut
tttttttt vxxxxxxu ??????? 326235224332212? ??????
4-20Performing White’s Test for
Heteroscedasticity
3,Obtain R2 from the auxiliary regression and multiply it by
the number of observations,T,It can be shown that
T R2 ??2 (m)
where m is the number of regressors in the auxiliary
regression excluding the constant term.
(还可以对辅助回归方程做 F 检验 )
4,If the ?2 test statistic from step 3 is greater than the
corresponding value from the statistical table then reject the
null hypothesis that the disturbances are homoscedastic.
? 例子,p150
4-21
Consequences of Using OLS in the
Presence of Heteroscedasticity
? OLS estimation still gives unbiased coefficient estimates,but
they are no longer BLUE.
? This implies that if we still use OLS in the presence of
heteroscedasticity,our standard errors could be inappropriate
and hence any inferences we make could be misleading.
? Whether the standard errors calculated using the usual
formulae are too big or too small will depend upon the form of
the heteroscedasticity.
4-22How Do we Deal with
Heteroscedasticity?
? If the form (i.e,the cause) of the heteroscedasticity is known,
then we can use an estimation method which takes this into
account (called generalised least squares,GLS).
? A simple illustration of GLS is as follows,Suppose that the
error variance is related to another variable zt by
? To remove the heteroscedasticity,divide the regression
equation by zt
Now
for known zt.
? ? 22v ar tt zu ??
t
t
t
t
t
tt
t v
z
x
z
x
zz
y ???? 3
3
2
21
1 ???
t
tt
z
uv ?
? ? ? ? 22 222v a rv a rv a r ?? ?????
?
?
???
??
t
t
t
t
t
t
t z
z
z
u
z
uv
4-23
Other Approaches to Dealing
with Heteroscedasticity
? So the disturbances from the new regression equation will be
homoscedastic.
? Other solutions (异方差的形式未知 ) include:
1,Transforming the variables into logs* or reducing by some
other measure of,size”.
2,Use White’s heteroscedasticity consistent相容 standard error
estimates.
The effect of using White’s correction is that in general the
standard errors for the slope coefficients are increased relative
to the usual OLS standard errors.
This makes us more,conservative” in hypothesis testing,so that
we would need more evidence against the null hypothesis before
we would reject it.
4-24
用 EViews检验和处理异方差
? White异方差检验
? p153
? White修正标准差估计
? p154-155
4-25
7 Assumption 3,cov(ui,uj) = 0 for i≠j
Background– The Concept of a Lagged Value
t yt yt-1 ?yt
1989M09 0.8 - -
1989M10 1.3 0.8 1.3-0.8 = 0.5
1989M11 -0.9 1.3 -0.9-1.3 = -2.2
1989M12 0.2 -0.9 0.2- -0.9= 1.1
1990M01 -1.7 0.2 -1.7-0.2 = -1.9
1990M02 2.3 -1.7 2.3- -1.7= 4.0
1990M03 0.1 2.3 0.1-2.3= -2.2
1990M04 0.0 0.1 0.0-0.1= -0.1
.,,,
.,,,
.,,,
4-26
Test for autocorrelation
? We assumed of the CLRM’s errors that Cov (ui,uj) = 0 for i?j,
i.e,This is essentially the same as saying there is no pattern in
the errors.
? Obviously we never have the actual u’s,so we use their sample
counterpart,the residuals (the ’s).
? If there are patterns in the residuals from a model,we say that
they are autocorrelated or serially correlated.
? Some stereotypical patterns we may find in the residuals are
given on the next 3 slides.
$ut
4-27
Positive Autocorrelation
Positive autocorrelation is indicated by a cyclical
residual plot over time.
+
-
-
t
u?
+
1
?
?t
u
- 3, 7
-6
- 6, 5
-6
- 3, 1
-5
-3
0, 5
-1
1
4
3
5
7
8
7
+
-
T i m e
t
u?
4-28
Negative Autocorrelation
Negative autocorrelation is indicated by an alternating pattern
where the residuals cross the time axis more frequently than
if they were distributed randomly
+
-
-
t
u?
+
1
?
?t
u
+
-
t
u?
T i m e
4-29
No pattern in residuals –
No autocorrelation
No pattern in residuals at all,this is what we would like to see
+
t
u?
-
-
+
1
?
?t
u
+
-
t
u?
4-30
Detecting Autocorrelation:
The Durbin-Watson Test
The Durbin-Watson (DW) is a test for first order
autocorrelation - i.e,it assumes that the relationship is between
an error and the previous one
ut = ?ut-1 + vt (1)
where vt ? N(0,?v2).
? The DW test statistic actually tests
H0, ? = 0 and H1, ?? 0
? The test statistic is calculated by
?
?有误
? ?
DW
u u
u
t t
t
T
t
t
T?
? ?
?
?
?
?
$ $
$
1
2
2
2
2
4-31
The Durbin-Watson Test,
Critical Values
? We can also write *
(2)
where is the estimated correlation coefficient,Since is a
correlation,it implies that,
? Rearranging for DW from (2) would give 0?DW?4.
If = 0,DW = 2,So roughly speaking,do not reject the null
hypothesis if DW is near 2 ? i.e,there is little evidence of
autocorrelation
? Unfortunately,DW has 2 critical values,an upper critical value
(du) and a lower critical value (dL),and there is also an
intermediate region where we can neither reject nor not reject
H0.
DW ? ?2 1( $ )?
$?
$?
1?1 ??? p
$?
4-32
The Durbin-Watson Test,Interpreting
the Results
Conditions which Must be Fulfilled for DW to be a Valid Test
1,Constant term in regression
2,Regressors are non-stochastic
3,No lags of dependent variable
4-33Another Test for Autocorrelation,
The Breusch-Godfrey Test
? It is a more general test for rth order autocorrelation:
?N(0,)
? The null and alternative hypotheses are:
H0, ?1 = 0 and ?2 = 0 and,.,and ?r = 0
H1, ?1 ? 0 or ?2 ? 0 or,.,or ?r ? 0
? The test is carried out as follows:
1,Estimate the linear regression using OLS and obtain the
residuals,.
2,Regress on all of the regressors from stage 1 (the x’s) plus
,Obtain R2 from this regression.
3,It can be shown that (T-r)R2 ??2(r)
$ut
$ut
u u u u u v vt t t t r t r t t? ? ? ? ? ?? ? ? ?? ? ? ?1 1 2 2 3 3,,,,
$,$,...,$u u ut t t r? ? ?1 2
2v?
4-34
The Breusch-Godfrey Test
? If the test statistic exceeds the critical value from the statistical
tables,reject the null hypothesis of no autocorrelation.
? The test is more general than the DW test.
? 这种检验的一个潜在困难是如何确定残差适当的滞后值 r。对
此没有明确的、一致的答案。可以对一组滞后值分别进行检
验，还常常考虑到使用数据的频率。例如，对于月度或季度
数据，滞后值 r可以取 12或 4。
4-35Consequences of Ignoring
Autocorrelation if it is Present
? The coefficient estimates derived using OLS are still unbiased,
but they are inefficient,i.e,they are not BLUE,even in large
sample sizes.
? Thus,if the standard error estimates are inappropriate,there
exists the possibility that we could make the wrong inferences.
? R2 is likely to be inflated relative to its,correct” value for
positively correlated residuals.
4-36
,Remedies” for Autocorrelation
? If the form of the autocorrelation is known,we could use a
GLS procedure – i.e,an approach that allows for
autocorrelated residuals e.g.,Cochrane-Orcutt procedure*.
? 在大样本情况下, 利用 D-W统计量, ?≈ 1－ d／ 2
? But such procedures that,correct” for autocorrelation require
assumptions about the form of the autocorrelation.
? If these assumptions are invalid,the cure would be more
dangerous than the disease! - see Hendry and Mizon (1978).
? However,it is unlikely to be the case that the form of the
autocorrelation is known,and a more,modern” view is that
residual autocorrelation presents an opportunity to modify the
regression.
? Newey-West 异方差相容自相关协方差估计
4-37
Dynamic Models
? All of the models we have considered so far have been static,
e.g.
yt = ?1 + ?2x2t +,.,+ ?kxkt + ut
? But we can easily extend this analysis to the case where the
current value of yt depends on previous values of y or one of
the x’s,e.g.(ADL)
yt = ?1 + ?2x2t +,.,+ ?kxkt + ?1yt-1 +?2x2t-1 + … + ?kxkt-1+ ut
? We could extend the model even further by adding extra lags,
e.g,x2t-2,yt-3,
? 加入多少滞后项是一个棘手的问题,相关理论可能有所帮助,
4-38
Models in First Difference Form
? Another way to sometimes deal with the problem of
autocorrelation is to switch to a model in first differences,
? Denote the first difference of yt,i.e,yt - yt-1 as ?yt; similarly for
the x-variables,?x2t = x2t - x2t-1 etc.
? The model would now be
?yt = ?1 + ?2 ?x2t +,.,+ ?k?xkt + ut
? Sometimes the change in y is purported to depend on previous
values of y or xt as well as changes in x:
?yt = ?1 + ?2 ?x2t + ?3x2t-1 +?4yt-1 + ut
4-39
产生自相关的原因
? 经济变量的惯性作用
– 消费刚性 ;销售量
? 经济行为的滞后性
– 基础设施建设周期 ;政策的滞后性质
? 随机因素的干扰
– 自然灾害、战争、政策的持续影响
? 设定偏误：所建模型不正确
– 模型函数形式不正确
– 遗漏主要变量
4-40Why Might we Want/Need To Include
Lags in a Regression?
? Inertia of the dependent variable
? Over-reactions(主要针对金融市场 )
? Other problems with the regression could cause the null
hypothesis of no autocorrelation to be rejected:
– Omission of relevant variables,which are themselves
autocorrelated.
– If we have committed a,misspecification” error by
using an inappropriate functional form.
– Autocorrelation resulting from unparameterised
seasonality.
4-41
The Long Run Static Equilibrium Solution
? One interesting property of a dynamic model is its long run or
static equilibrium solution.
?,Equilibrium” implies that the variables have reached some
steady state and are no longer changing,i.e,if y and x are in
equilibrium,we can say
yt = yt+1 =,.,=y and xt = xt+1 =,.,=x
Consequently,?yt = yt - yt-1 = y - y = 0 etc.
? So the way to obtain a long run static solution is:
1,Remove all time subscripts from variables
2,Set error terms equal to their expected values,E(ut)=0
3,Remove first difference terms altogether
4,Gather terms in x together and gather terms in y together.
4-42
The Long Run Static Equilibrium
Solution,An Example
? If our model is
?yt = ?1 + ?2 ?x2t + ?3 x2t-1 +?4 yt-1 + ut
then the static solution would be given by
0 = ?1 + ?3 x2 +?4 y
?4 y = - ?1 - ?3 x2
2
4
3
4
1 xy
?
?
?
? ???
4-43Problems with Adding Lagged
Regressors to,Cure” Autocorrelation
? Inclusion of lagged values of the dependent variable violates
the assumption that the RHS variables are non-stochastic.
在小样本情况下, 将因变量的滞后变量作为解释变量可能导
致参数估计是有偏的 （ 但仍然是一致的 ） 。
? What does an equation with a large number of lags actually
mean?
– 包含滞后或差分项的模型可能难以解释并且无法用于检验
原先的理论 。
? Note that if there is still autocorrelation in the residuals of a
model including lags,then the OLS estimators will not even be
consistent.
4-44
8 假设 4,Xt 是非随机变量
? 可以证明：当存在随机的回归自变量时，只要自变量
与扰动项是不相关的,OLS估计量仍然是无偏的和一
致的。
– 证明见第 178页。
? 如果自变量 与扰动项相关，其处理方法将在第六章
中讨论。
4-45
9 Assumption 5：扰动项服从正态分布
? Why did we need to assume normality for hypothesis testing?
Testing for Departures from Normality
? The Bera 贝拉 Jarque 雅克 normality test
? Skewness and kurtosis are the (standardised) third and fourth
moments of a distribution.
? A normal distribution is not skewed and is defined to have a
coefficient of kurtosis of 3.
? The kurtosis of the normal distribution is 3 so its excess
kurtosis is zero.
4-46
Normal versus Skewed Distributions
A normal distribution A skewed distribution
f( x )
x
x
f( x )
4-47Leptokurtic尖峰 versus Normal
Distribution
-5, 4 -3, 6 -1, 8 -0, 0 1, 8 3, 6 5, 4
0, 0
0, 1
0, 2
0, 3
0, 4
0, 5
4-48
Testing for Normality
? Bera and Jarque formalise this by testing the residuals for
normality by testing whether the coefficient of skewness and the
coefficient of excess kurtosis are jointly zero.
? It can be proved that the coefficients of skewness and kurtosis
can be expressed respectively as:
and
? The Bera Jarque test statistic is given by
? We estimate b1 and b2 using the residuals from the OLS
regression,.*$u
? ?b
E u
1
3
2 3 2
? [ ]/
? ? ?
b E u2
4
2 2
? [ ]
?
? ? ? ?2~
24
3
6
2
2
2
2
1 ??
?
?
?
?
? ??? bbTW
4-49What do we do if we find evidence of
Non-Normality?
? It is not obvious what we should do!
? Could use a method which does not assume normality,but
difficult and what are its properties?
? 当样本容量充分大时, 扰动项的正态性假定不太重要 。
? Often the case that one or two very extreme residuals
causes us to reject the normality assumption.
? An alternative is to use dummy variables.
e.g,say we estimate a monthly model of asset returns from
1980-1990,and we plot the residuals,and find a
particularly large outlier for October 1987:
4-50What do we do if we find evidence
of Non-Normality?
? Create a new variable:
D87M10t = 1 during October 1987 and zero otherwise.
This effectively knocks out that observation,But we need a
theoretical reason for adding dummy variables,*
O c t
1987
+
-
T i m e
t
u?
4-51
10 Multicollinearity
? This problem occurs when the explanatory variables are very
highly correlated with each other.
? Perfect multicollinearity
- e.g,suppose x3 = 2x2,Cannot estimate all the coefficients
? Near Multicollinearity
? Problems if Near Multicollinearity is Present but Ignored
- R2 will be high but the individual coefficients will have high
standard errors.
- The regression becomes very sensitive to small changes in
the specification.
- Thus confidence intervals for the parameters will be very
wide,and significance tests might therefore give
inappropriate conclusions.
4-52
产生多重共线性的可能原因
? 许多经济变量往往存在共同的变化趋势。
– 如经济增长、收入增长、产品销售增长、储蓄增长
? 用截面数据建立回归模型，解释变量常常从经济意义上存在
着密切的关联度。
– 如资本、劳动力、能源、科技进步等都与企业的生产规模密切相关。
? 在模型中较多采用滞后变量
– 滞后变量从经济性质上看与原来的变量并没有区别。
? 解释变量选择不当
– 如化肥使用量、农业生产资金投入同时作为解释变量
4-53
Measuring Multicollinearity
? The easiest way to measure the extent of multicollinearity is
simply to look at the matrix of correlations between the
individual variables,e.g.
? 变量显著性 （ t） 与方程显著性 (R2,F)的综合判断
? But another problem,if 3 or more variables are linear
e.g,x2t + x3t = x4t,it would be very difficult to detect.
? Note that high correlation between y and one of the x’s is not
muticollinearity.
C orr x 2 x 3 x 4
x 2 - 0,2 0,8
x 3 0,2 - 0,3
x 4 0,8 0,3 -
4-54
Solutions to the Problem of
Multicollinearity
? 多重共线性不影响 OLS估计的 BLUE性质 。
?, Traditional” approaches,such as ridge regression 岭回归 or
principal components,But these usually bring more problems
than they solve.
? Some econometricians argue that if the model is otherwise OK,
just ignore it。 特别是用于预测时 。
? The easiest ways to,cure” the problems are
- drop one of the collinear variables。 但须考虑理论的要求 。
- transform the highly correlated variables into a ratio*
- go out and collect more data 增加样本容量 。 e.g.
- a longer run of data
- switch to a higher frequency
Use pooled sample*
? 逐步回归法 *
4-55
11 Adopting the Wrong Functional Form
? We have previously assumed that the appropriate functional
form is linear,This may not always be true.
? We can formally test this using Ramsey’s RESET test
（ regression specification error test ）,which is a
general test for mis-specification of functional form.
? Essentially the method works by adding higher order terms of
the fitted values (e.g,etc.) into an auxiliary regression:
Regress on powers of the fitted values:
The test statistic is given by TR2 and is distributed as
a,
? So if the value of the test statistic is greater than a then
reject the null hypothesis that the functional form was correct.
$ $ $,,, $u y y y vt t t p t p t? ? ? ? ? ??? ? ? ?0 1 2 2 3 1
$,$y yt t2 3
?2 1( )p ?
$ut
?2 1( )p ?
4-56
What do we do if this is the case?
? The RESET test gives us no guide as to what a better
specification might be.
? 对于参数非线性模型, 一般需要使用非线性估计方法 。
? 对于一些参数线性而解释变量非线性的模型, 仍可以使用
OLS进行估计 。
? 例如 if the true model is
In this case the remedy is obvious.
? Another possibility is to transform the data into logarithms.
This will linearise many previously multiplicative models into
additive ones:
ttttt uxxxy ????? 324223221 ????
tttutt uxyeAxy t ????? lnln ???
4-5712 Omission of an Important Variable
13 Inclusion of an Irrelevant Variable
Omission of an Important Variable
? Consequence,The estimated coefficients on all the other
variables will be biased and inconsistent unless the
excluded variable is uncorrelated with all the included
variables.
? Even if this condition is satisfied,the estimate of the
coefficient on the constant term will be biased.
? The standard errors will also be biased.
Inclusion of an Irrelevant Variable
? Coefficient estimates will still be consistent and unbiased,
but the estimators will be inefficient.
4-58
14 Parameter Stability Tests
? So far,we have estimated regressions such as
? We have implicitly assumed that the parameters (?1,?2 and ?3)
are constant for the entire sample period.
? We can test this implicit assumption using parameter stability
tests,The idea is essentially to split the data into sub-periods
and then to estimate up to three models,for each of the sub-
parts and for all the data and then to,compare” the RSS of the
models.
? There are two types of test we can look at:
- Chow test (analysis of variance test)
- Predictive failure tests
y t = ? 1 + ? 2 x 2 t + ? 3 x 3 t + u t
4-59
The Chow邹至庄 Test
? The steps involved are:
1,Split the data into two sub-periods,Estimate the
regression over the whole period and then for the two sub-
periods separately (3 regressions),Obtain the RSS for each
regression.
2,The restricted regression is now the regression for the
whole period while the,unrestricted regression” comes in
two parts,for each of the sub-samples.
We can thus form an F-test which is the difference between
the RSS’s,The statistic is
? ?R S S R S S R S S
R S S R S S
T k
k
? ?
? ?
?1 2
1 2
2
4-60
The Chow邹至庄 Test
RSS = RSS for whole sample
RSS1 = RSS for sub-sample 1
RSS2 = RSS for sub-sample 2
T = number of observations
2k = number of regressors in the,unrestricted” regression
(since it comes in two parts)
k = number of regressors in (each part of the),unrestricted”
regression
? 3,Perform the test,If the value of the test statistic is greater
than the critical value from the F-distribution,which is an
F(k,T-2k),then reject the null hypothesis that the
parameters are stable over time.*
4-61
A Chow Test Example
? Consider the following regression for the CAPM ? (again) for
the returns on Glaxo.
? Say that we are interested in estimating Beta for monthly data
from 1981-1992,The model for each sub-period is
? 1981M1 - 1987M10
Rgt =0.24 + 1.2RMt T = 82 RSS1 = 0.03555
? 1987M11 - 1992M12
Rgt = 0.68 + 1.53RMt T = 62 RSS2 = 0.00336
? 1981M1 - 1992M12
Rgt = 0.39 + 1.37RMt T = 144 RSS = 0.0434
4-62
A Chow Test Example - Results
? The null hypothesis is
?
= 7.698
? Compare with 5% F(2,140) = 3.06
We reject H0 at the 5% level and say that we reject the
restriction that the coefficients are the same in the two
periods.
H and0 1 2 1 2,? ? ? ?? ?
? ?T e s t s t a t i s t i c ? ? ?
? ?
?0 0434 0 0355 0 00336
0 0355 0 00336
144 4
2
.,,
.,
4-63
The Predictive Failure Test
? Problem with the Chow test is that we need to have enough data
to do the regression on both sub-samples,i.e,T1>>k,T2>>k.
? An alternative formulation is the predictive failure test.
? What we do with the predictive failure test is estimate the
regression over a,long” sub-period (i.e,most of the data) and
then we predict values for the other period and compare the two.
- Run the regression for the whole period (the restricted
regression) and obtain the RSS
- Run the regression for the,large” sub-period and obtain the
RSS (called RSS1),Note we call the number of observations T1
where T2 = number of observations we are attempting to
“predict”,The test statistic will follow an F(T2,T1-k).*
2
1
1
1S t at i s t i cT es t
T
kT
R S S
R S SR S S ????
4-64
Backwards versus Forwards
Predictive Failure Tests
? There are 2 types of predictive failure tests:
- Forward predictive failure tests,where we keep the last few
observations back for forecast testing,e.g,we have
observations for 1970Q1-1994Q4,So estimate the model over
1970Q1-1993Q4 and forecast 1994Q1-1994Q4.
- Backward predictive failure tests,where we attempt to
“back-cast” the first few observations,e.g,if we have data for
1970Q1-1994Q4,and we estimate the model over 1971Q1-
1994Q4 and backcast 1970Q1-1970Q4.
4-65
Predictive Failure Tests – An Example
? We have the following models estimated:
For the CAPM ? on Glaxo(!).
? 1981M1-1992M12
Rgt = 0.39 + 1.37RMt T = 144 RSS = 0.0434
? 1981M1-1990M12
Rgt = 0.32 + 1.31RMt T1 = 120 RSS1 = 0.0420
Can this regression adequately,forecast” the values?
= 0.164
Compare with F(24,118) = 1.66.
So we do not reject the null hypothesis that the model can
adequately predict the observations for the last two years,
24
21 2 0
0 4 2 0.0
0 4 2 0.00 4 3 4.0S t a t i s t i cT e s t ????
4-66
How do we decide the sub-parts to use?
0
200
400
600
800
1000
1200
1400
1
27 53 79
1
0
5
1
3
1
1
5
7
1
8
3
2
0
9
2
3
5
2
6
1
2
8
7
3
1
3
3
3
9
3
6
5
3
9
1
4
1
7
4
4
3
S a m p l e P e ri o d
V
a
l
u
e
o
f
S
e
r
i
e
s
(
y
t
)
? As a rule of thumb,we could use all or some of the following:
- Plot the dependent variable over time and split the data
accordingly to any obvious structural changes in the series,e.g.
4-67
How do we decide the sub-parts to use?
- Split the data according to any known important historical
events (e.g,stock market crash,new government elected)
- Use all but the last few observations and do a forward
predictive failure test on those.
- Use all but the first few observations and do a backward
predictive failure test on those.
? 如果参数 稳定性检验被拒绝, 则可采用
– 重新设定模型, 如引入其它变量
– 对子样本分别进行参数估计
? Eviews中的稳定性检验（参见 p206-207）
4-68
15 A Strategy for Building
Econometric Models
Our Objective:
? To build a statistically adequate empirical model which
- satisfies the assumptions of the CLRM
- is parsimonious
- has the appropriate theoretical interpretation
- has the right,shape” - i.e.
- all signs on coefficients are,correct”
- all sizes of coefficients are,correct”
- is capable of explaining the results of all competing
models
4-6915 Approaches to Building
Econometric Models
? There are 2 popular philosophies of building econometric models,the
“specific-to-general” and,general-to-specific” approaches.
?,Specific-to-general” was used almost universally until the mid 1980’s,
and involved starting with the simplest model and gradually adding to it.
? Little,if any,diagnostic testing was undertaken,But this meant that all
inferences were potentially invalid.
? 如果初始模型被错误设定, 诊断检验也不一定能可靠指出问题的来源 。
? An alternative and more modern approach to model building is the
“LSE” or Hendry,general-to-specific” methodology.
? The advantages of this approach are that it is statistically sensible and
also the theory on which the models are based usually has nothing to say
about the lag structure of a model.因此,模型的滞后结构主要由数据本
身来确定 。 此外, 更注重考虑遗漏相关变量的后果 。
? 这种方法认为, 一个好的模型与数据和理论都是一致的；而且应包含
竞争 rival模型 。 建议广泛使用诊断检验以保证模型的统计准确性 。
4-70
Hendry提出的模型标准
最终的可接受模型应当满足以下一些标准：
? 逻辑上合理
? 与理论一致，包括满足相关的参数约束
? 回归自变量与误差项不相关
? 参数估计值在整个样本区间上是稳定的
? 残差是白噪声（完全随机的）
? 能够解释所有竞争模型的结果，并给出更多的解释 (包容原则 )
– 简约包容 parsimonious encompassing：小模型能解释大模型的所有
结果
4-71
The General-to-Specific Approach
? First step is to form a,large” model with lots of variables on the
right hand side。 GUM (generalised unrestricted model)
? At this stage,we want to make sure that the model satisfies all of
the assumptions of the CLRM
? If the assumptions are violated,we need to take appropriate
actions to remedy this,e.g.
- taking logs
- adding lags
- dummy variables
? We need to do this before hypotheses test
? Once we have a model which satisfies the assumptions,it could
be very big with lots of lags & independent variables
4-72
The General-to-Specific Approach:
Reparameterising the Model
? The next stage is to reparameterise the model by
- knocking out very insignificant regressors
- some coefficients may be insignificantly different from
each other,so we can combine them.
? At each stage,we need to check the assumptions are still OK.
? Hopefully at this stage,we have a statistically adequate
empirical model which we can use for
- testing underlying financial theories
- forecasting future values of the dependent variable
- formulating policies,etc.
4-73
16 Regression Analysis Example:
Determinants of Sovereign Credit Ratings
? Cantor and Packer (1996)
Financial background:
? What are sovereign credit ratings and why are we interested in them?
? Two ratings agencies (Moody’s and Standard and Poor’s) provide credit
ratings for many governments.
? Each possible rating is denoted by a grading:
Moody’s Standard and Poor’s
Aaa AAA
…… …,.
B3 B-
4-74
Purposes of the Paper
- to attempt to explain and model how the ratings agencies arrived at
their ratings.
- to use the same factors to explain the spreads of sovereign yields
above a risk-free proxy
- to determine what factors affect how the sovereign yields react to
ratings announcements
4-75
Determinants of Sovereign Ratings
? Data
Quantifying the ratings (dependent variable),Aaa/AAA=16,...,B3/B-=1
? Explanatory variables (units of measurement):
- Per capita income in 1994 (thousands of dollars)
- Average annual GDP growth 1991-1994 (%)
- Average annual inflation 1992-1994 (%)
- Fiscal balance,Average annual government budget surplus as a
proportion of GDP 1992-1994 (%)
- External balance,Average annual current account surplus as a proportion
of GDP 1992-1994 (%)
- External debt Foreign currency debt as a proportion of exports 1994 (%)
- Dummy for economic development
- Dummy for default history
Income and inflation are transformed to their logarithms.
4-76
The model,Linear and estimated using OLS
Dep en d en t V ar i ab le
Expl an at o r y V a r i a bl e
Expec t ed
si g n
Av er ag e
R at i ng
M ood y ’ s
R at i ng
S& P
R at i ng
M ood y ’ s / S& P
Dif f er e nce
I nt er cep t? 1.442
( 0.663 )
3.408
( 1.379 )
- 0.524
( - 0.223 )
3.932* *
( 2.521 )
Per c api t a i nco m e + 1.242** *
( 5.302 )
1.027** *
( 4.041 )
1.458 ***
( 6.048 )
- 0.431 ***
( - 2.688 )
GD P g r owt h + 0.151
( 1.935 )
0.130
( 1.545 )
0.171 **
( 2.132 )
- 0.040
( 0.756 )
I nf l at i on - - 0.611 ***
( - 2.839 )
- 0.630 ***
( - 2.701 )
- 0.591 ***
( 2.671 )
- 0.039
( - 0.265 )
Fi sc a l B a l an ce + 0.073
( 1.324 )
0.049
( 0.818 )
0.097 *
( 1.71 )
- 0.048
( - 1.274 )
Ext er nal B al anc e + 0.003
( 0.314 )
0.006
( 0.535 )
0.001
( 0.046 )
0.006
( 0.779 )
Ext er nal D ebt - - 0.013 ***
( - 5.088 )
- 0.015 ***
( - 5.365 )
- 0.011 ***
( - 4.236 )
- 0.004 ***
( - 2.133 )
Dev el opm ent dum m y + 2.7 76 * **
( 4.25 )
2,957* * *
( 4.175 )
2.595 ***
( 3.861 )
0.362
( 0.81 )
Def au l t d um m y - - 2.042 ***
( - 3.175 )
- 1.63* *
( - 2.097 )
- 2.622 ***
( - 3.962 )
1.159** *
( 2.632 )
Adj us t ed R
2
0.924 0.905 0.926 0.836
No tes,t - r atio s i n p ar en t h e s es ; *,* *,an d * * * in d ica te s i g n i f ica n ce at t h e 1 0 %,5 % a n d 1 % lev el s
r esp ec tiv el y, So u r ce, Ca n to r an d P ac k er ( 1 9 9 6 ), Re p r in ted w it h p er m i s s io n f r o m I n s t itu ti o n a l I n ve s to r,
4-77
Interpreting the Model
From a statistical perspective
? Virtually no diagnostics
? Adjusted R2 is high
? Look at the residuals,actual rating - fitted rating
From a financial perspective
? Do the coefficients have their expected signs and sizes?
Do Ratings Add to Publicly Available Information?
? Now dependent variable is
- Log (Yield on the sovereign bond - yield on a US treasury bond)
4-78
Do Ratings Add to Publicly Available Available
Information? Results
De pe nde nt V a r iable,L o g ( y ield s p r e a d )
Va r iable Ex pe c ted S ig n ( 1) ( 2) ( 3)
I nte r c e pt? 2.105***
( 16.148)
0.466
( 0.345)
0.074
( 0.071)
Ave r a ge
R a ti ng
- - 0.221***
( - 19.175)
- 0.218***
( - 4.276)
Per c api t a
i nco m e
- - 0.144
( - 0.927)
0.226
( 1.523)
GD P g r owt h - - 0.004
( - 0.142)
0.029
( 1.227)
I nf l at i on + 0.108
( 1.393)
- 0.004
( - 0.068)
Fi sc a l B a l an ce - - 0.037
( - 1.557)
- 0.02
( - 1.045)
Ext er nal
B al anc e
- - 0.038
( - 1.29)
- 0.023
( - 1.008)
Ext er nal D ebt + 0.003***
( 2.651)
0.000
( 0.095)
Dev el opm ent
dumm y
- - 0.723***
( - 2.059)
- 0.38
( - 1.341)
Def au l t d um m y + 0.612***
( 2.577)
0.085
( 0.385)
Adj us t ed R
2
0.919 0.857 0.914
No tes,t - r atio s i n p ar en t h e s es ; *,* *,an d * * * in d ica te s i g n i f ica n ce at t h e 1 0 %,5 % a n d 1 % lev el s
r esp ec tiv el y, So u r ce, Ca n to r an d P ac k er ( 1 9 9 6 ), Re p r in ted w it h p er m i s s io n f r o m I n s t itu ti o n a l I n ve s to r,
4-79
What Determines How the Market Reacts
to Ratings Announcements?
? The sample,Every announcement of a ratings change that occurred between
1987 and 1994 - 79 such announcements spread over 18 countries.
? 39 were actual ratings changes
? 40 were,watchlist / outlook”changes
? The dependent variable,changes in the relative spreads over the US T-bond
over a 2-day period at the time of the announcement.
4-80
What Determines How the Market Reacts
to Ratings Announcements? Explanatory variables.
0 /1 dummies for
- Whether the announcement was positive
- Whether there was an actual ratings change
- Whether the bond was speculative grade
- Whether there had been another ratings announcement in the previous 60 days.
and
- The change in the spread over the previous 60 days.
- The ratings gap between the announcing and the other agency
4-81
What Determines How the Market Reacts
to Ratings Announcements? Results
D e pe nde nt V a ria ble, L o g Re la ti ve Spre a d
I nd e pe nde nt v a ria ble Coef fic ie nt ( t - ra ti o)
I nte rcept - 0.02
( - 1.4)
P osit ive a nnounc e me nts 0.01
(0,34)
Rating s c ha n ge s - 0.01
( - 0.37)
Mood y ’s a nnounc e m e nts 0.02
(1,51)
S pe c ula ti ve g ra d e 0.03**
(2,33)
Chang e in re la ti ve spr e a d s fr om da y – 60 to da y - 1 - 0.06
( - 1.1)
Rating g a p 0.03*
(1,7)
O the r ra ti n g a nnoun c e me nts fr om da y – 60 to da y - 1 0.05**
(2,15)
A djust e d R
2
0.12
No te,* an d * * d en o te s i g n i f i can ce at t h e 1 0 % an d 5 % le v els r esp ectiv el y, S o u r ce,C an t o r an d P ack e r
( 1 9 9 6 ), R e p r in ted w ith p er m is s io n f r o m I n s titu tio n a l I n vesto r,
4-82
Conclusions
? 6 factors appear to play a big role in determining sovereign credit ratings -
incomes,GDP growth,inflation,external debt,industrialised or not,and default
history.
? The ratings provide more information on yields than all of the macro factors put
together.
? We cannot determine well what factors influence how the markets will react to
ratings announcements.
4-83
Comments on the Paper
? Only 49 observations for first set of regressions and 35 for yield regressions and
up to 10 regressors
? No attempt at reparameterisation
? Little attempt at diagnostic checking
? Where did the factors (explanatory variables) come from?
