10.3 Time Series Thus Far

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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

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Assumption TS.4(Homoskedasticity)
Conditional on X, the variance of ut is the same
for all t
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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

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Assumption TS.4 Violation

• Consider the regression

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
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Assumption TS.5(No Serial Correlation)
Conditional on X, errors in two different time
periods are uncorrelated
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Assumption TS.5 Notes

• If we assume that X is non-random, TS.5
  simplifies to

-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 didnt need
this assumption for cross-sectional data as
random sampling ensured no connection between
error terms
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Assumption TS.5 Violation

• Take the regression

If actual weight is unexpectedly high one time
period (high fat intake), then utgt0, and weight
can be expected to be high in subsequent periods
(ut1gt0) Likewise if weight is unexpectedly low
one time period (liposuction), then utlt0, and
weight can be expected to be low in subsequent
periods (ut1lt0)
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10.3 Gauss Markov Assumptions

• -Assumptions TS.1 through TS. 5 are our
  Gauss-Markov 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

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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
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
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Theorem 10.3(Unbiased Estimation of s2)
Under assumptions TS.1 through TS.5, the
estimator
Is an unbiased estimator of s2, where dfn-k-1
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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
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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

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Assumption TS.6(Normality)
The errors ut are independent of X and are
independently and identically distributed as
Normal (0, s2).
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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.
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10.4 Time Series Logs

• -Logarithms used in time series regressions again
  refer to percentage changes

-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
(delta0delta1 in this case) is called the
LONG-RUN ELASTICITY -measuring change in
utility 2 periods after a 1 increase in sleep
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10.4 Time Series Dummy Variables

• -Time series data can benefit from dummy
  variables much like time series data
• -DVs can indicate when a characteristic changes
• -ie Rain1 days that it rains
• -DVs 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

-Where Exams1 during exams
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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

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10.4 Index Review

• -An index is generally equal to 100 in the base
  year. Base years are changed using

-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
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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

-To see if the TV event made an impact, test if
delta00 -one could also include and test
multiplicative Dummy Variables
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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

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10.5 Linear Time Trend

• -The linear time trend is a simple model of
  trending

-Where et is an independent, identically
distributed sequence with E(et)0 and
Var(et)se2 -the change in y between any two
periods is equal to alpha1 -if alpha1gt0, y is
growing over time and has an upward trend -if
alpha1gt0, y is growing over time and has an
upward trend
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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

-Here each periods change in log(yt) is equal to
alpha1 -As weve seen previously, if growth is
small, the percentage growth rate of yt each
period is equal to 100(alpha1)
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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

-Using derivatives, here the one-period increase
in yt is shown as
-Although more complicated trends are possible,
they run the risk of explaining variation that
should be attributed to x and not t
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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

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10.5 Spurious Regressions
-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
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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

-In the above example, y has been linearly
detrended using the regression
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10.5 Detrending

• 2) Run the following regression. Intercepts are
  not needed, and will be estimated as zero if not
  omitted

-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
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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 ys trending
• -the simplest solution is to calculate R2 from a
  regression where y has been detrended

-Note that only the y has been detrended and t is
included as an explanatory variable
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10.5 R2 and Trending

• This R2 can be calculated as

-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
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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

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10.5 Deseasonalizing

• Just as data can be deseasonalized, it can also
  be detrended
• Regress each y and x variable on seasonal dummy
  variables and obtain the residuals

• 2) Regress the deseasonalized (residuals) y on
  the deseasonalized xs

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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.