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Econometrics
Basic Regression analysis with
Time Series Data
Anna Donina
Lecture 9
The nature of time series data
2
• Temporal ordering of observations; may not be arbitrarily reordered
• Typical features: serial correlation/nonindependence of
observations
• How should we think about the randomness in time series data?
• The outcome of economic variables (e.g. GNP, Dow Jones) is
uncertain; they should therefore be modeled as random
variables
• Time series are sequences of r.v. (= stochastic processes)
• Randomness does not come from sampling from a population
• „Sample“ = the one realized path of the time series out of the
many possible paths the stochastic process could have taken
The nature of time series data
3
• Example: US inflation and unemployment rates 1948-2003
Here, there are only two time series.
There may be many more variables
whose paths over time are observed
simultaneously.
Time series analysis focuses on
modeling the dependency of a variable
on its own past, and on the present and
past values of other variables.
Examples of Time Series Regression Models
4
• Static models
• In static time series models, the current value of one variable is
modeled as the result of the current values of explanatory
variables
• Examples for static models
There is a contemporaneous
relationship between
unemployment and inflation (=
Phillips-Curve).
The current murder rate is determined by the current
conviction rate, unemployment rate, and fraction of young
males in the population.
Examples of Time Series Regression Models
5
• Finite distributed lag models
• In finite distributed lag models, the explanatory variables are
allowed to influence the dependent variable with a time lag
• Example for a finite distributed lag model
• The fertility rate may depend on the tax value of a child, but
for biological and behavioral reasons, the effect may have a lag
Children born per
1,000 women in
year t
Tax exemption
in year t
Tax exemption
in year t-1
Tax exemption
in year t-2
Examples of Time Series Regression Models
6
• Interpretation of the effects in finite distributed lag models
• Effect of a past shock on the current value of the dep. variable
Effect of a transitory shock:
If there is a one time shock in
a past period, the dep.
variable will change
temporarily by the amount
indicated by the coefficient
of the corresponding lag.
Effect of permanent shock:
If there is a permanent shock in a past
period, i.e. the explanatory variable
permanently increases by one unit, the
effect on the dep. variable will be the
cumulated effect of all relevant lags. This is
a long-run effect on the dependent variable.
7
Finite sample properties of OLS under classical
assumptions
• Assumption TS.1 (Linear in parameters)
• Assumption TS.2 (No perfect collinearity)
„In the sample (and therefore in the underlying time
series process), no independent variable is constant
nor a perfect linear combination of the others.“
The time series involved obey a linear relationship. The stochastic
processes yt, xt1,…, xtk are observed, the error process ut is
unobserved. The definition of the explanatory variables is general,
e.g. they may be lags or functions of other explanatory variables.
8
Finite sample properties of OLS under classical
assumptions
• Notation
• Assumption TS.3 (Zero conditional mean)
The mean value of the unobserved factors is
unrelated to the values of the explanatory
variables in all periods
The values of all explanatory
variables in period number t
This matrix collects all the
information on the complete
time paths of all explanatory
variables
9
Finite sample properties of OLS under classical
assumptions
• Discussion of assumption TS.3
• Strict exogeneity is stronger than contemporaneous exogeneity
• TS.3 rules out feedback from the dep. variable on future
values of the explanatory variables; this is often questionable
esp. if explanatory variables „adjust“ to past changes in the
dependent variable (example: murder rate and police pc)
• If the error term is related to past values of the explanatory
variables, one should include these values as
contemporaneous regressors
The mean of the error term is unrelated to
the values of the explanatory variables of
all periods
The mean of the error term is unrelated to
the explanatory variables of the same
period
Exogeneity:
Strict exogeneity:
10
Finite sample properties of OLS under classical
assumptions
• Theorem 10.1 (Unbiasedness of OLS)
• Assumption TS.4 (Homoscedasticity)
• A sufficient condition is that the volatility of the error is
independent of the explanatory variables and that it is
constant over time
• In the time series context, homoscedasticity may also be easily
violated, e.g. if the volatility of the dep. variable depends on
regime changes (example: T-bill rate and infl, deficit)
The volatility of the errors must not be
related to the explanatory variables in
any of the periods
11
Finite sample properties of OLS under classical
assumptions
• Assumption TS.5 (No serial correlation)
• Discussion of assumption TS.5
• The assumption may easily be violated if, conditional on
knowing the values of the indep. variables, omitted factors are
correlated over time
• Why was such an assumption not made in the cross-sectional
case?
• The assumption may also serve as substitute for the random
sampling assumption if sampling a cross-section is not done
completely randomly
• In this case, given the values of the explanatory variables,
errors have to be uncorrelated across cross-sectional units (e.g.
states)
Conditional on the explanatory variables,
the unobserved factors must not be
correlated over time
12
Finite sample properties of OLS under classical
assumptions
• Theorem 10.2 (OLS sampling variances)
• Theorem 10.3 (Unbiased estimation of the error variance)
Under assumptions TS.1 – TS.5:
The same formula as in
the cross-sectional case
13
Finite sample properties of OLS under classical
assumptions
• Theorem 10.4 (Gauss-Markov Theorem)
• Under assumptions TS.1 – TS.5, the OLS estimators have the
minimal variance of all linear unbiased estimators of the
regression coefficients
• This holds conditional as well as unconditional on the
regressors
• Assumption TS.6 (Normality)
• Theorem 10.5 (Normal sampling distributions)
• Under assumptions TS.1 – TS.6, the OLS estimators have the
usual normal distribution (conditional on x).
The usual F- and t-tests are valid.
independently of x
This assumption implies TS.3 – TS.5
14
Finite sample properties of OLS under classical
assumptions
• Example: Static Phillips curve
• Discussion of CLM assumptions
Contrary to theory, the estimated
Phillips Curve does not suggest a
tradeoff between inflation and
unemployment
A linear relationship might be restrictive, but it
should be a good approximation. Perfect collinearity
is not a problem as long as unemployment varies over
time.
TS.1:
The error term contains factors
such as monetary shocks,
income/demand shocks, oil price
shocks, supply shocks, or
exchange rate shocks
TS.2:
15
Finite sample properties of OLS under classical
assumptions
• Discussion of CLM assumptions (cont.)
TS.3:
For example, past unemployment shocks may lead to future
demand shocks which may dampen inflation
For example, an oil price shock means more inflation and
may lead to future increases in unemployment
TS.4:
TS.5:
Assumption is violated if monetary policy
is more „nervous“ in times of high
unemployment
TS.6:
Assumption is violated if exchange
rate influences persist over time
(they cannot be explained by
unemployment)
Questionable
Easily violated
16
Finite sample properties of OLS under classical
assumptions
• Example: Effects of inflation and deficits on interest rates
• Discussion of CLM assumptions
A linear relationship might be restrictive, but it
should be a good approximation. Perfect collinearity
will seldomly be a problem in practice.
TS.1:
The error term represents other
factors that determine interest
rates in general, e.g. business
cycle effects
TS.2:
Interest rate on 3-months T-bill Government deficit as percentage of
GDP
17
Finite sample properties of OLS under classical
assumptions
• Discussion of CLM assumptions (cont.)
TS.3:
For example, past deficit spending may boost economic activity, which in
turn may lead to general interest rate rises
For example, unobserved demand shocks may increase interest rates and
lead to higher inflation in future periods
TS.4:
TS.5:
Assumption is violated if higher deficits lead to more
uncertainty about state finances and possibly more
abrupt rate changes
TS.6:
Assumption is violated if business cylce effects
persist across years (and they cannot be
completely accounted for by inflation and the
evolution of deficits)
Questionable
Easily violated
18
Using Dummy Explanatory Variables in Time
Series
• Interpretation
• During World War II, the fertility rate was temporarily lower
• It has been permanently lower since the introduction of the
pill in 1963
Children born per 1,000
women in year t
Tax
exemption in
year t
Dummy for World
War II years (1941-45)
Dummy for availabity of
contraceptive pill (1963-present)
19
Example for a time
series with a linear
upward trend
Time series with trends
20
Time series with trends
• Modelling a linear time trend
• Modelling an exponential time trend
Abstracting from random deviations, the dependent
variable increases by a constant amount per time unit
Alternatively, the expected value of the
dependent variable is a linear function of time
Abstracting from random deviations, the
dependent variable increases by a constant
percentage per time unit
21
• Example for a time series with an exponential trend
Abstracting from random
deviations, the time
series has a constant
growth rate
Time series with trends
22
Time series with trends
• Using trending variables in regression analysis
• If trending variables are regressed on each other, a spurious
relationship may arise if the variables are driven by a common
trend
• In this case, it is important to include a trend in the regression
• Example: Housing investment and prices
Per capita housing
investment
Housing price
index
It looks as if investment and
prices are positively related
23
Time series with trends
• Example: Housing investment and prices (cont.)
• When should a trend be included?
• If the dependent variable displays an obvious trending behaviour
• If both the dependent and some independent variables have
trends
• If only some of the independent variables have trends; their effect
on the dep. var. may only be visible after a trend has been
substracted
There is no significant
relationship between price and
investment anymore
24
Time series with trends
• A Detrending interpretation of regressions with a time trend
• It turns out that the OLS coefficients in a regression including a
trend are the same as the coefficients in a regression without a
trend but where all the variables have been detrended before the
regression
• This follows from the general interpretation of multiple
regressions
• Computing R-squared when the dependent variable is trending
• Due to the trend, the variance of the dep. var. will be overstated
• It is better to first detrend the dep. var. and then run the
regression on all the indep. variables (plus a trend if they are
trending as well)
• The R-squared of this regression is a more adequate measure of fit
25
Time series with trends
• Modelling seasonality in time series
• A simple method is to include a set of seasonal dummies:
• Similar remarks apply as in the case of deterministic time trends
• The regression coefficients on the explanatory variables can be
seen as the result of first deseasonalizing the dependent and
the explanatory variables
• An R-squared that is based on first deseasonalizing the
dependent variable may better reflect the explanatory power
of the explanatory variables
=1 if obs. from
december
=0 otherwise