ec3090 - econometrics - week 10 - time series econometrics

EC3090 - EconometricsWeek 10 - Time Series Econometrics

Agustın S. Benetrix

Week 10 - December 2015

Agustın Benetrix (TCD) EC3090 - Econometrics Week 10 - December 2015 1 / 54

Plan for the coming weeks

1 Basic Regression Analysis with Time Series Data

2 Further issues of using OLS with Time Series Data

3 Serial Correlation and Heteroskedasticity in Time Series Regressions

4 Advanced Time Series Topics


Basic regression analysis with time series data

The nature of time series data

• Temporal ordering of observations; may not be arbitrarily reordered

• Typical features: serial correlation /nonindependence of observations



How should we think about therandomness in time series data?

• The outcome of economic variables (e.g. GNP, Dow Jones) isuncertain; they should therefore be modelled 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 manypossible paths the stochastic process could have taken



Figure : US inflation and unemployment rates 1948-2003

• Time series analysis focuses on modelling the dependency of a variableon its own past, and on the present and past values of other variables


Basic regression analysis with time series data Examples of time series regression models

Static models

In static time series models, the current value of one variable is modeled asthe result of the current values of explanatory variables

Example

yt = β0 + β1zt + ut t = 1, 2, ....., n

Example

There is a contemporaneous relationship between unemployment andinflation (= Phillips-Curve)

inft = β0 + β1unemt + ut



Static models

Example

The current murder rate is determined by the current conviction rate,unemployment rate, and fraction of young males in the population

mrdrtet = β0 + β1convrtet + β2unemt + β3yngmlet + ut



Finite distributed lag models (FDL)

In finite distributed lag models, the explanatory variables are allowed toinfluence the dependent variable with a time lag

Example

The fertility rate may depend on the tax value of a child, but for biologicaland behavioral reasons, the effect may have a lag

gfrt = α0 + δ0pet + δ1pet-1 + δ2pet-2 + ut




In general we have..

yt = α0 + δ0zt + δ1zt−1 + δ2zt−2 + ut

This is a FDL model of order two



Finite distributed lag models (FDL)To interpret the coefficients above suppose:

• z is a constant and it is equal to c in all periods before t

• It then increases by 1 unit in period t and then reverts back to itsprecious level in t + 1..

.., zt−2 = c, zt−1 = c , zt = c + 1, zt+1 = c , zt+2 = c

If we set the error term in each period to be zero we get

yt−1 = α0 + δ0c + δ1c + δ2c

yt = α0 + δ0 (c + 1) + δ1c + δ2c

yt+1 = α0 + δ0c + δ1 (c + 1) + δ2c

yt+2 = α0 + δ0c + δ1c + δ2 (c + 1)

yt+3 = α0 + δ0c + δ1c + δ2c




From the first two equations we know that

yt − yt−1 = δ0

This shows that δ0 is the immediate change in y due to one-unit increasein z at time t

B δ0 is the impact multiplier




If we want to see what the effect of a permanent increase in z is:

yt−1 = α0 + δ0c + δ1c + δ2c

yt = α0 + δ0 (c + 1) + δ1c + δ2c

yt+1 = α0 + δ0 (c + 1) + δ1 (c + 1) + δ2c

yt+2 = α0 + δ0 (c + 1) + δ1 (c + 1) + δ2 (c + 1)

• With a permanent increase in z , after one period y has increased byδ0 + δ1

• After two periods it increases by δ0 + δ1 + δ2 and so on

B δ0 + δ1 + δ2 is called long-run multiplier




• The effect is biggest after a lag of one period. After that, the effectvanishes (if the initial shock was transitory)

• The long run effect of a permanent shock is the cumulated effect ofall relevant lagged effects. It does not vanish (if the initial shock is apermanent one)




A finite distributed lag model of order q can be written as follows

yt = α0 + δ0zt + δ1zt−1 + ..+ δqzt−q + ut

With the long-run multiplier being

LRP = δ0 + δ1 + ..+ δq


Basic regression analysis with time series data Finite sample properties of OLS under CLM assumptions

Finite sample properties of OLS underCLM assumptions

Unbiasedness of OLS

Assumption TS1 [Linear in parameters]: The stochastic process{(xt1, xt2, .....xtk , yt) : t = 1, 2, ....., n} follows the linear model

yt = β0 + β1xt1 + ..+ βkxtk + ut

where {ut : t = 1, 2, ...., n} is the sequence of errors or disturbances. Heren is the number of observations (time periods).

• The time series involved obey a linear relationship

• The stochastic processes yt , xt1, xt2, .....xtk are observed, the errorprocess ut is unobserved

• The definition of the explanatory variables is general• e.g. they may be lags or functions of other explanatory variables.




Unbiasedness of OLS

Assumption TS2 [Zero conditional mean]: For each t, the expectedvalue of the error ut , given the explanatory variables for all time periods, iszero.

E (ut |X) = 0, t = 1, 2, ...n

• This assumption implies that the error at time t, ut , is uncorrelatedwith each explanatory variable in every time period

• The mean value of the unobserved factors is unrelated to the valuesof the explanatory variables in all periods




Unbiasedness of OLS

Definition (Exogeneity)

E (ut |xt) = 0

• The mean of the error term is unrelated to the explanatory variablesof the same period

Definition (Strict exogeneity)

E (ut |X) = 0

This is stronger than contemporaneous exogeneity

• The mean of the error term is unrelated to the values of theexplanatory variables of all periodsAgustın Benetrix (TCD) EC3090 - Econometrics Week 10 - December 2015 17 / 54



Unbiasedness of OLS

• TS2 rules out feedback from the dependent variable on future valuesof the explanatory variables; this is often questionable especially ifexplanatory variables adjust to past changes in the dependent variable

• If the error term is related to past values of the explanatory variables,one should include these values as contemporaneous regressors




Unbiasedness of OLS

Notation

X =

x11 x12 · · · x1k...

......

xt1 xt2 · · · xtk...

......

xn1 xn2 · · · xnk

• This matrix collects all the information on the complete time paths of

all explanatory variables

• The values of all explanatory variables in period number t




Unbiasedness of OLS

Assumption TS3 [No perfect collinearity]: In the sample (and thereforein the underlying time series process), no independent variable is constantor a perfect linear combination of the others

• In the sample (and therefore in the underlying time series process), noindependent variable is constant nor a perfect linear combination ofthe others.”




Unbiasedness of OLS

Theorem (10.1 Unbiasedness of OLS)

Under assumptions TS1, TS2 and TS3, the OLS estimators are unbiasedconditional on X, and therefore unconditionally as well

E (βj) = βj , j = 0, 1, .....k




The variances of the OLS estimators

Assumption TS4 [Homoskedasticity]: Conditional on X, the variance ofut is the same for all t

Var(ut |X) = Var(ut) = σ2, , t = 1, 2, ....n




The variances of the OLS estimators

• Assumption TS4 means that Var(ut |X) cannot depend on X

• The volatility of the errors must not be related to the explanatoryvariables in any of the periods

• A sufficient condition is that the volatility of the error is independentof the explanatory variables and that it is constant over time

• In the time series context, homoscedasticity may also be easilyviolated, e.g. if the volatility of the dependent variable depends onregime changes

• When TS4 does not hold, we say that the errors are heteroskedastic,just as in the cross-sectional case




Assumption TS5 [No serial correlation]: Conditional on X, the errors intwo different time periods are uncorrelated

Corr(ut , us |X) = 0 for all s 6= t

• Conditional on the explanatory variables, the un-observed factorsmust not be correlated over time



The easiest way to think of TS5 assumption is to ignore the conditioningon X. Then, Assumption TS5 is simply

Corr(ut , us) = 0 for all s 6= t

When the above expression is false, we say that the errors in

yt = β0 + β1xt1 + ..+ βkxtk + ut

suffer from serial correlation, or autocorrelation, because they arecorrelated across time



• Why was such an assumption not made in the cross-sectional case?

• The assumption may easily be violated if, conditional on knowing thevalues of the independent variables, omitted factors are correlatedover time

• In the cross-section case, given the values of the explanatory variables,errors have to be uncorrelated across cross-sectional units (e.g. states)



Theorem (10.2 OLS sampling variances)

Under the series of Gauss-Markov assumptions, the variance of βjconditional on X is

Var(βj |X) =σ2

SSTj(1− R2j ), j = 1, ...., k

where SSTj is the total sum of squares of xtj and R2j is the R-squared

from the regressions of xj on the other independent variables

• The same formula as in the cross-sectional case

• The conditioning on the values of the explanatory variables effectivelymeans that, in a finite sample, one ignores the sampling variabilitycoming from the randomness of the regressors



Theorem (10.3 Unbiased estimation of σ2)

Under assumptions TS1-TS5, the estimator σ2 = SSRdf is and unbiased

estimator of σ2 ,where df = n − k − 1



Gauss-Markov theorem

Theorem (10.4 Gauss-Markov)

Under assumptions TS1-TS5, the OLS estimators are the best linearunbiased estimators conditional on X.



Gauss-Markov theorem

• The OLS estimators have the minimal variance of all linear unbiasedestimators of the regression coefficients

• This holds conditional as well as unconditional on the regressors



Inference under the Classical Linear Model(CLM) assumptions

Assumption TS6 [Normality]: The errors ut are independent of X andare independent and identically distributed (i.i.d.) as N(0, σ2)



Inference under the CLM assumptions

Theorem (10.5 Normal sampling distribution)

Under assumptions TS1-TS6 (the Classical Linear Model or CLMassumptions), the OLS estimators are normally distributed, conditional onX. Further, under the null hypothesis, each t-statistic has a t distribution,and each F statistic has an F distribution. The usual construction ofconfidence intervals is also valid.



Inference under the CLM assumptionsStatic Phillips curve example

Example (Static Phillips curve)

inft = 1.42 + 0.468unemt

(1.72) (0.289)

n = 49,R2 = 0.053,R2

= 0.033

Contrary to theory, the estimated Phillips Curve does not suggest atradeoff between inflation and unemployment



Discussion of the CLM assumptionsStatic Phillips curve example

TS1: In inft = β0 + β1unemt + ut the error term contains factors such asmonetary shocks, income/demand shocks, oil price shocks, supply shocks,or exchange rate shocks

TS2: E (ut |unem1, ...,unemn) = 0 is easily violated

• For example, past unemployment shocks may lead to future demandshocks which may dampen inflation: unemt-1 ↑→ ut ↓

• For example, an oil price shock means more inflation and may lead tofuture increases in unemployment: ut−1 ↑→unemt ↑

TS3: A linear relationship might be restrictive, but it should be a goodapproximation. Perfect collinearity is not a problem as long asunemployment varies over time



Discussion of the CLM assumptionsStatic Phillips curve example

TS4: Var(ut |unem1, ...,unemn) = σ2 Assumption is violated if monetarypolicy is more ”nervous” in times of high unemployment

TS5: Corr(ut , us |unem1, ...,unemn) = 0 assumption is violated ifex-change rate influences persist over time (they cannot be explained byunemployment)

TS6: ut ∼ N(0, σ2) is a questionable assumption in the context of model



Inference under the CLM assumptionsEffects of inflation and deficits on interest rates example

Example (Effects of inflation and deficits on interest rates)

i3t = 1.73 + 0.606inft + 0.606deft

(0.43) (.082) (0.118)

n = 56,R2 = 0.602,R2

= 0.587

• Variables: Interest rate on 3-months T-bill, inflation rate andGovernment deficit as percentage of GDP




TS1: In i3t = β0 + β1inft + β1deft + ut the error term represents otherfactors that determine interest rates in general, e.g. business cycle effects

TS2: E (ut |inf1, ...,infn,def1, ...,defn) = 0 is easily violated

• For example, past deficit spending may boost economic activity,which in turn may lead to general interest rate rises: deft-1 ↑→ ut ↑

• For example, unobserved demand shocks may increase interest ratesand lead to higher inflation in future periods: ut ↑→deft-1 ↑

TS3: A linear relationship might be restrictive, but it should be a goodapproximation. Perfect collinearity will seldomly be a problem in practice




TS4: Assumption is violated if higher deficits lead to more uncertaintyabout state finances and possibly more abrupt rate changesVar(ut |inf1, ...,infn,def1, ...,defn) = σ2

TS5: Corr(ut , us |inf1, ...,infn,def1, ...,defn) = 0 assumption is violated ifbusiness cycle effects persist across years (and they cannot be completelyaccounted for by inflation and the evolution of deficits)

TS6: ut ∼ N(0, σ2) is a questionable assumption in the context of model


Trends and Seasonality

Trending time series



Trending time seriesWhat kind of statistical models adequately capture trending

behaviour?

• One popular formulation is to write the series {yt} as

yt = α0 + α1t + et , t = 1, 2, ...

where, in the simplest case, {et} is an independent, identically distributed(i.i.d.) sequence with E (et) = 0, Var(et) = σ2

e

• Note how the parameter α1 multiplies time,t, resulting in a lineartime trend

• Interpretation: α1 measures the change in yt from one period to thenext due to the passage of time

∆yt = yt − yt−1 = α1




Another way to think about a sequence that has a linear time trend is thatits average value is a linear function of time:

E (yt) = α0 + α1t



Trending time seriesMany economic time series are better approximated by an exponentialtrend.

• NB: Abstracting from random deviations, the time series has aconstant growth rate




• We model an exponential time trend as follows

ln(yt) = β0 + β1t + et , t = 1, 2, ..

• How do we interpret β1?

∆ ln(yt) ≈yt − yt−1

yt

• The r.h.s. is the growth rate of output

• It follows then that if ∆et = 0 then

∆ ln(yt) = β1 for all t

• Abstracting from random deviations, the dependent variable increasesby a constant percentage per time unit, ie constant growth rate




• Although linear and exponential trends are the most common, timetrends can be more complicated. For example, we might have aquadratic time trend

yt = α0 + α1t + α2t2 + et , t = 1, 2, ...

• If α1 and α2 are positive, then the slope of the trend is increasing, asis easily seen by computing the approximate slope (holding et fixed):

∆yt∆t≈ α1 + 2α2t



Using Trending Variables in RegressionAnalysis

• Accounting for explained or explanatory variables that are trending isfairly straightforward in regression analysis

• Nothing about trending variables necessarily violates the classicallinear model assumptions

• However, we must be careful to allow for the fact that unobserved,trending factors that affect yt might also be correlated with theexplanatory variables




• If we ignore this possibility, we may find a spurious relationshipbetween yt and one or more explanatory variables

• The phenomenon of finding a relationship between two or moretrending variables simply because each is growing over time is anexample of spurious regression

• Fortunately, adding a time trend eliminates this problem

yt = β0 + β1xt1 + β2xt2 + β3t + ut




Example (Housing investment and prices)

ln(invpc) = −.550 + 1.241 ln(price)

(.043) (.382)

n = 42,R2 = .208,R2

= .189

Variables: Per capita housing investment and Housing price index

• It looks as if investment and prices are positively related




Example (Housing investment and prices)

Now we include a linear trend

ln(invpc) = −.913 + .381 ln(price) + .0098t

(.136) (.679) (.0035)

n = 42,R2 = .341,R2

= .307

• There is no significant relationship between price and investmentanymore



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 onthe dependent variable may only be visible after a trend has beensubstracted



Computing R2

• Due to the trend, the variance of the dependent varariable will beoverstated

• It is better to first detrend the dependent variable and then run theregression on all the independent variables (plus a trend if they aretrending as well)

• The R2 of this regression is a more adequate measure of fit


Seasonality

Seasonality

Generally, we can include a set of seasonal dummy variables to account forseasonality in the dependent variable, the independent variables, or both

yt = β0 + δ1febt + δ2mart + δ3aprt + δ4mayt + ..+ β1xt1 + ...+ βkxtk + ut

• Just as including a time trend in a regression has the interpretation ofinitially detrending the data, including seasonal dummies in aregression can be interpreted as deseasonalizing the data


Seasonality

Seasonality

• If a time series is observed at monthly or quarterly intervals (or evenweekly or daily), it may exhibit seasonality

• For example, monthly housing starts in the Midwest are stronglyinfluenced by weather. While weather patterns are somewhat random,we can be sure that the weather during January will usually be moreinclement than in June, and so housing starts are generally higher inJune than in January

• One way to model this phenomenon is to allow the expected value ofthe series, yt , to be different in each month


Seasonality

Summary

1. We have covered basic regression analysis with time series data

2. Under assumptions that parallel those for cross-sectional analysis,OLS is unbiased (under TS1 through TS3), OLS is BLUE (under TS1through TS5), and the usual OLS standard errors, t statistics, and Fstatistics can be used for statistical inference (under TS1 through TS6)

3. Because of the temporal correlation in most time series data, wemust explicitly make assumptions about how the errors are related to theexplanatory variables in all time periods and about the temporalcorrelation in the errors themselves


Seasonality

Summary

5. The classical linear model assumptions can be pretty restrictive fortime series applications, but they are a natural starting point. We haveapplied them to both static regression and finite distributed lag models

6. Trends and seasonality can be easily handled in a multipleregression framework by including time and seasonal dummy variables


ec3090 - econometrics - week 10 - time series econometrics

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