10.3 Time Series Thus Far
Whereas cross sectional data needed 3
assumptions to make OLS unbiased, time
series data needs only 2
-Although the third assumption is much
stronger
-If we omit a valid variable, we cause biased as
seen and calculated in Chapter 3
-Now all that remains is to derive assumptions
that allow us to test the significance of our
OLS estimates
Assumption TS.4
(Homoskedasticity)
Conditional on X, the variance of ut
is the same for all t:
Var (ut | X )  Var (ut )   , t  1,2,..., n.
2
Assumption TS.4 Notes
-essentially, the variance of the error term cannot
depend on X; it must be constant
-it is sufficient if:
1) ut and X are independent
2) Var (ut) is constant over time
-ie: no trending
-if TS.4 is violated we again have
heteroskedasticity
-Chapter 12 shows similar tests for Het as found
in Chapter 8
Assumption TS.4 Violation
Consider the regression:
tuitiont   0  1inflation t  politics t  ut
Unfortunately, tuition is often a political rather
than an economic decision, leading to tuition
freezes (=real tuition decreases) in an
attempt to buy votes
-This effect can span time periods
-Since politics can affect the variability of tuition,
this regression is heteroskedastic
Assumption TS.5
(No Serial Correlation)
Conditional on X, errors in two
different time periods are
uncorrelated:
Cor (ut , us | X )  0 t s
Assumption TS.5 Notes
If we assume that X is non-random, TS.5
simplifies to:
Cor (ut , us )  0 ts
(10.12)
-If this assumption is violated, we say that our
time series errors suffer from
AUTOCORRELATION, as they are correlated
across time
-note that TS.5 assumes nothing about
intertemporal correlation among x variables
-we didn’t need this assumption for crosssectional data as random sampling ensured
no connection between error terms
Assumption TS.5 Violation
Take the regression:
weightt   0  1calories t  exerciset  ut
If actual weight is unexpectedly high one time
period (high fat intake), then ut>0, and
weight can be expected to be high in
subsequent periods (ut+1>0)
Likewise if weight is unexpectedly low one time
period (liposuction), then ut<0, and weight
can be expected to be low in subsequent
periods (ut+1<0)
10.3 Gauss Markov Assumptions
-Assumptions TS.1 through TS. 5 are our GaussMarkov assumptions for time series data
-They allow us to estimate OLS variance
-If cross sectional data is not random, TS.1
through TS.5 can sometimes be used in cross
sectional applications
-with these 5 properties in time series data, we
see variance calculated and the Gauss-Markov
theorem holding the same as with cross
sectional data
-the same OLS properties apply in finite sample
time series as in cross-sectional data:
Theorem 10.2
(OLS Sampling Variances)
Under the time series Gauss-Markov Assumptions
TS.1 through TS.5, the variance of Bjhat,
conditional on X, is
Var ( ˆ j | X ) 

2
SST j (1  R )
2
j
, j  0, 1,..., k.
Where SSTj is the total sum of squares of xtj and
Rj2 is the R-squared from the regression of xj on
the other independent variables
Theorem 10.3
2
(Unbiased Estimation of σ )
Under assumptions TS.1 through TS.5,
the estimator
SSR
ˆ 
n  k 1
2
Is an unbiased estimator of σ2, where
df=n-k-1
Theorem 10.4
(Gauss-Markov Theorem)
Under assumptions TS.1 through
TS.5, the OLS estimators are the
best linear unbiased estimators
conditional on X
10.3 Time Series and Testing
-In order to construct valid standard errors, t
statistics and F statistics, we need to add one
more assumption
-TS.6 implies and is stronger than TS.3, TS.4 and
TS.5
-given these 6 time series assumptions, tests are
conducted identically to the cross sectional
case
-time series assumptions are more restrictive
than cross sectional assumptions
Assumption TS.6
(Normality)
The errors ut are independent of X
and are independently and
identically distributed as Normal
(0, σ2).
Theorem 10.5
(Normal Sampling Distribution)
Under assumptions TS.1 through TS.6,
the CLM assumptions for time series,
the OLS estimators are normally
distributed, conditional on X. Further,
under the null hypothesis, each t
statistic has a t distribution, and each F
statistic has an F distribution. The
usual construction of confidence
intervals is also valid.
10.4 Time Series Logs
-Logarithms used in time series regressions again
refer to percentage changes:
log( U t )   0   0 log( sleept )  1 log( sleept 1 )  otherfactorst  ut
-here the impact propensity, delta0 is also called
the SHORT-RUN ELASTICITY
-it measures the immediate percentage
change in utility given a 1% increase in sleep
-the long-run propensity (delta0+delta1 in this
case) is called the LONG-RUN ELASTICITY
-measuring change in utility 2 periods after
a 1% increase in sleep
10.4 Time Series Dummy Variables
-Time series data can benefit from dummy
variables much like time series data
-DV’s can indicate when a characteristic changes
-ie: Rain=1 days that it rains
-DV’s can also refer to periods of time to see if
there are systematic differences between time
periods
-for example, if you suspect base utility to be
different during exams:
U t   0   0 Exams  otherfactorst  ut
-Where Exams=1 during exams
10.4 Index Review
-an index number aggregates a vast amount of
information into a single quantity
-for example, Econ 399 time can be spent in
class, reviewing the text/notes, studying,
working on assignments, or working on your
paper
-since all these individual factors are highly
correlated (an one hour in one area is not
necessarily the same as one hour elsewhere)
and numerous to conclude, work on Econ 399
can instead be shown as an index
10.4 Index Review
-An index is generally equal to 100 in the base
year. Base years are changed using:
old index t
new index t 
100
old index new base
-where old indexnew base is the old value of the
index in the new base year
-a special case of indexes is a price index, which
is also useful to convert to REAL variables:
nominal
real 
100
PI
10.4 Index Review
-indexes and Dummy Variables can be used
together for event studies; to test if an event
has a structural impact on a regression:
Your favorite character on TV is killed off, and
you want to test if this affects your econ 399
performance. You estimate the regression:
Mark t   0   0Char  1Work  otherfactorst  ut
-To see if the TV event made an impact, test if
delta0=0
-one could also include and test multiplicative
Dummy Variables
10.5 Time Trends
-Sometimes economic data has a TIME TREND; a
tendency to grow over time
-if two variables are either increasing or
decreasing over time, they will appear to be
correlated although they may be independent
-failure to account for trending can lead to errors
in a regression
-even one variable trending in a regression can
lead to errors, as we shall see
10.5 Linear Time Trend
-The linear time trend is a simple model of
trending: yt   0  1t  et , t  1, 2, ..., n (10.24)
-Where et is an independent, identically
distributed sequence with E(et)=0 and
Var(et)=σe2
-the change in y between any two periods is
equal to alpha1
-if alpha1>0, y is growing over time and has an
upward trend
-if alpha1>0, y is growing over time and has an
upward trend
10.5 Exponential Time Trend
-The linear time trend allows for the same
increase in y every period
-An exponential time trend allows for the same
PERCENTAGE increase in y each period:
log( yt )   0  1t  et , t  1, 2, ..., n (10.26)
-Here each period’s change in log(yt) is equal to
alpha1
-As we’ve seen previously, if growth is small, the
percentage growth rate of yt each period is equal
to 100(alpha1)%
10.5 Quadratic Time Trend
-While linear and exponential time trends are
most common, more complicated trends can
occur
-For example, take a quadratic time trend:
yt   0  1t   2t 2  et
(10.29)
-Using derivatives, here the one-period increase
in yt is shown as:
yt
 1  2 2t (10.30)
t
-Although more complicated trends are possible,
they run the risk of explaining variation that
should be attributed to x and not t
10.5 Spurious Regressions
-Trending variables do not themselves cause a
violation of TS.1 through TS.6
-however, if y and at least one x variable appear
to be correlated due to trending, the regression
suffers from a SPURIOUS REGRESSION
PROBLEM
-if y itself is trending, we have the true
regression:
yt   0  1 xt1   2 xt 2  3t  ut
10.5 Spurious Regressions
yt   0  1 xt1   2 xt 2  3t  ut
-If we omit the valid “variable” t, we have caused
bias
-this effect is heightened if x variables are also
trending
-adding a time trend can actually make a variable
more significant if its movement about its trend
affects y
-note that including a time trend is also valid if
only x (and not y) is trending
10.5 Detrending
-Including a time trend can be seen as similar to
“partialling out” the trending of variables:
1) Regress y and all x variables on the time trend
and save the residuals such that:
yt  yt  ˆ 0  ˆ1t
-In the above example, y has been linearly
detrended using the regression:
yt  ˆ 0  ˆ1t  et
10.5 Detrending
2) Run the following regression. Intercepts are
not needed, and will be estimated as zero if not
omitted: y   x   x  ...   x  v
t
1 t1
2 t2
k tk
-These betas will be identical to the regression
with a time trend included
-this shows why including a time trend is also
important if x is trending; the OLS estimates are
still affected by the trend
10.5 R2 and Trending
Typical R2 for time series regressions is artificially
high as SST/(n-1) is no longer an unbiased or
consistent estimator in the face of trending
-R2 cannot account for y’s trending
-the simplest solution is to calculate R2 from a
regression where y has been detrended:
yt   0  1 xt1   2 xt 2  ...   k xtk   0t  v
-Note that only the y has been detrended and t is
included as an explanatory variable
10.5 R2 and Trending
This R2 can be calculated as:
SSR
R  1
2
 yt
2
-Note that SSR is the same for both the models
with and without t
-this R2 will always be lower than or equal to the
typical R2
-this R2 can be adjusted to account for variable
inclusion
-when doing F tests, the typical R2 is still used
10.5 Seasonality
-Some data may exhibit SEASONALITY, it may
naturally vary within the year; within seasons
-ie: housing starts, ice cream sales
-typically data that exhibits seasonal patterns is
seasonally adjusted
-if this is not the case, seasonal dummy variables
should be included (11 montly dummy variables,
3 seasonal dummy variables, etc)
-significance tests can then be performed to
evaluate the seasonality of the data
10.5 Deseasonalizing
Just as data can be deseasonalized, it can also be
detrended:
1) Regress each y and x variable on seasonal
dummy variables and obtain the residuals:
yt  yt  ˆ 0  ˆ1Spring  ˆ 2 Summer  ˆ 3 Fall
2) Regress the deseasonalized (residuals) y on
the deseasonalized x’s:
yt  1xt1   2 xt 2  ...   k xtk  v
10.5 Deseasonalizing
This deseasonalized model is again a better
source for accurate R2 values
-as this model nets out any variation
attributed to seasonality
-Note that some regressions may suffer from
both trending and seasonality, requiring both
detrending and deseasonalizing, which
requires including seasonal dummy variables
and a time trend in step 1 above.