Economics 20 - Prof,Anderson 1
Time Series Data
yt = b0 + b1xt1 +,,,+ bkxtk + ut
1,Basic Analysis
Economics 20 - Prof,Anderson 2
Time Series vs,Cross Sectional
Time series data has a temporal ordering,
unlike cross-section data
Will need to alter some of our assumptions
to take into account that we no longer have
a random sample of individuals
Instead,we have one realization of a
stochastic (i.e,random) process
Economics 20 - Prof,Anderson 3
Examples of Time Series Models
A static model relates contemporaneous
variables,yt = b0 + b1zt + ut
A finite distributed lag (FDL) model allows
one or more variables to affect y with a lag,
yt = a0 + d0zt + d1zt-1 + d2zt-2 + ut
More generally,a finite distributed lag
model of order q will include q lags of z
Economics 20 - Prof,Anderson 4
Finite Distributed Lag Models
We can call d0 the impact propensity – it
reflects the immediate change in y
For a temporary,1-period change,y returns
to its original level in period q+1
We can call d0 + d1 +…+ dq the long-run
propensity (LRP) – it reflects the long-run
change in y after a permanent change
Economics 20 - Prof,Anderson 5
Assumptions for Unbiasedness
Still assume a model that is linear in
parameters,yt = b0 + b1xt1 +,,,+ bkxtk + ut
Still need to make a zero conditional mean
assumption,E(ut|X) = 0,t = 1,2,…,n
Note that this implies the error term in any
given period is uncorrelated with the
explanatory variables in all time periods
Economics 20 - Prof,Anderson 6
Assumptions (continued)
This zero conditional mean assumption
implies the x’s are strictly exogenous
An alternative assumption,more parallel to
the cross-sectional case,is E(ut|xt) = 0
This assumption would imply the x’s are
contemporaneously exogenous
Contemporaneous exogeneity will only be
sufficient in large samples
Economics 20 - Prof,Anderson 7
Assumptions (continued)
Still need to assume that no x is constant,
and that there is no perfect collinearity
Note we have skipped the assumption of a
random sample
The key impact of the random sample
assumption is that each ui is independent
Our strict exogeneity assumption takes care
of it in this case
Economics 20 - Prof,Anderson 8
Unbiasedness of OLS
Based on these 3 assumptions,when using
time-series data,the OLS estimators are
unbiased
Thus,just as was the case with cross-
section data,under the appropriate
conditions OLS is unbiased
Omitted variable bias can be analyzed in
the same manner as in the cross-section case
Economics 20 - Prof,Anderson 9
Variances of OLS Estimators
Just as in the cross-section case,we need to
add an assumption of homoskedasticity in
order to be able to derive variances
Now we assume Var(ut|X) = Var(ut) = s2
Thus,the error variance is independent of
all the x’s,and it is constant over time
We also need the assumption of no serial
correlation,Corr(ut,us| X)=0 for t ? s
Economics 20 - Prof,Anderson 10
OLS Variances (continued)
Under these 5 assumptions,the OLS
variances in the time-series case are the
same as in the cross-section case,Also,
The estimator of s2 is the same
OLS remains BLUE
With the additional assumption of normal
errors,inference is the same
Economics 20 - Prof,Anderson 11
Trending Time Series
Economic time series often have a trend
Just because 2 series are trending together,
we can’t assume that the relation is causal
Often,both will be trending because of
other unobserved factors
Even if those factors are unobserved,we
can control for them by directly controlling
for the trend
Economics 20 - Prof,Anderson 12
Trends (continued)
One possibility is a linear trend,which can
be modeled as yt = a0 + a1t + et,t = 1,2,…
Another possibility is an exponential trend,
which can be modeled as log(yt) = a0 + a1t
+ et,t = 1,2,…
Another possibility is a quadratic trend,
which can be modeled as yt = a0 + a1t +
a2t2 + et,t = 1,2,…
Economics 20 - Prof,Anderson 13
Detrending
Adding a linear trend term to a regression
is the same thing as using,detrended”
series in a regression
Detrending a series involves regressing
each variable in the model on t
The residuals form the detrended series
Basically,the trend has been partialled out
Economics 20 - Prof,Anderson 14
Detrending (continued)
An advantage to actually detrending the
data (vs,adding a trend) involves the
calculation of goodness of fit
Time-series regressions tend to have very
high R2,as the trend is well explained
The R2 from a regression on detrended data
better reflects how well the xt’s explain yt
Economics 20 - Prof,Anderson 15
Seasonality
Often time-series data exhibits some
periodicity,referred to seasonality
Example,Quarterly data on retail sales
will tend to jump up in the 4th quarter
Seasonality can be dealt with by adding a
set of seasonal dummies
As with trends,the series can be seasonally
adjusted before running the regression
Time Series Data
yt = b0 + b1xt1 +,,,+ bkxtk + ut
1,Basic Analysis
Economics 20 - Prof,Anderson 2
Time Series vs,Cross Sectional
Time series data has a temporal ordering,
unlike cross-section data
Will need to alter some of our assumptions
to take into account that we no longer have
a random sample of individuals
Instead,we have one realization of a
stochastic (i.e,random) process
Economics 20 - Prof,Anderson 3
Examples of Time Series Models
A static model relates contemporaneous
variables,yt = b0 + b1zt + ut
A finite distributed lag (FDL) model allows
one or more variables to affect y with a lag,
yt = a0 + d0zt + d1zt-1 + d2zt-2 + ut
More generally,a finite distributed lag
model of order q will include q lags of z
Economics 20 - Prof,Anderson 4
Finite Distributed Lag Models
We can call d0 the impact propensity – it
reflects the immediate change in y
For a temporary,1-period change,y returns
to its original level in period q+1
We can call d0 + d1 +…+ dq the long-run
propensity (LRP) – it reflects the long-run
change in y after a permanent change
Economics 20 - Prof,Anderson 5
Assumptions for Unbiasedness
Still assume a model that is linear in
parameters,yt = b0 + b1xt1 +,,,+ bkxtk + ut
Still need to make a zero conditional mean
assumption,E(ut|X) = 0,t = 1,2,…,n
Note that this implies the error term in any
given period is uncorrelated with the
explanatory variables in all time periods
Economics 20 - Prof,Anderson 6
Assumptions (continued)
This zero conditional mean assumption
implies the x’s are strictly exogenous
An alternative assumption,more parallel to
the cross-sectional case,is E(ut|xt) = 0
This assumption would imply the x’s are
contemporaneously exogenous
Contemporaneous exogeneity will only be
sufficient in large samples
Economics 20 - Prof,Anderson 7
Assumptions (continued)
Still need to assume that no x is constant,
and that there is no perfect collinearity
Note we have skipped the assumption of a
random sample
The key impact of the random sample
assumption is that each ui is independent
Our strict exogeneity assumption takes care
of it in this case
Economics 20 - Prof,Anderson 8
Unbiasedness of OLS
Based on these 3 assumptions,when using
time-series data,the OLS estimators are
unbiased
Thus,just as was the case with cross-
section data,under the appropriate
conditions OLS is unbiased
Omitted variable bias can be analyzed in
the same manner as in the cross-section case
Economics 20 - Prof,Anderson 9
Variances of OLS Estimators
Just as in the cross-section case,we need to
add an assumption of homoskedasticity in
order to be able to derive variances
Now we assume Var(ut|X) = Var(ut) = s2
Thus,the error variance is independent of
all the x’s,and it is constant over time
We also need the assumption of no serial
correlation,Corr(ut,us| X)=0 for t ? s
Economics 20 - Prof,Anderson 10
OLS Variances (continued)
Under these 5 assumptions,the OLS
variances in the time-series case are the
same as in the cross-section case,Also,
The estimator of s2 is the same
OLS remains BLUE
With the additional assumption of normal
errors,inference is the same
Economics 20 - Prof,Anderson 11
Trending Time Series
Economic time series often have a trend
Just because 2 series are trending together,
we can’t assume that the relation is causal
Often,both will be trending because of
other unobserved factors
Even if those factors are unobserved,we
can control for them by directly controlling
for the trend
Economics 20 - Prof,Anderson 12
Trends (continued)
One possibility is a linear trend,which can
be modeled as yt = a0 + a1t + et,t = 1,2,…
Another possibility is an exponential trend,
which can be modeled as log(yt) = a0 + a1t
+ et,t = 1,2,…
Another possibility is a quadratic trend,
which can be modeled as yt = a0 + a1t +
a2t2 + et,t = 1,2,…
Economics 20 - Prof,Anderson 13
Detrending
Adding a linear trend term to a regression
is the same thing as using,detrended”
series in a regression
Detrending a series involves regressing
each variable in the model on t
The residuals form the detrended series
Basically,the trend has been partialled out
Economics 20 - Prof,Anderson 14
Detrending (continued)
An advantage to actually detrending the
data (vs,adding a trend) involves the
calculation of goodness of fit
Time-series regressions tend to have very
high R2,as the trend is well explained
The R2 from a regression on detrended data
better reflects how well the xt’s explain yt
Economics 20 - Prof,Anderson 15
Seasonality
Often time-series data exhibits some
periodicity,referred to seasonality
Example,Quarterly data on retail sales
will tend to jump up in the 4th quarter
Seasonality can be dealt with by adding a
set of seasonal dummies
As with trends,the series can be seasonally
adjusted before running the regression
