ec3090 - econometrics - time series econometrics · 2015. 12. 4. · time series econometrics agust...

EC3090 - EconometricsTime Series Econometrics

Agustın S. Benetrix

Week 11 - December 2015

Agustın Benetrix (TCD) EC3090 - Econometrics Week 11 - December 2015 1 / 50

Further issues of using OLS with TS data Stationarity

Stationarity

• A stationary time series process is one whose probability distributionsare stable over time

• If we take any collection of random variables in the sequence and thenshift that sequence ahead h time periods, the joint probabilitydistribution must remain unchanged



Stationary stochastic process

Definition (Stationary Stochastic Process)

The stochastic process {xt : t = 1, 2, ....} is stationary if for everycollection of time indices 1 ≤ t1 ≤ t2... ≤ tm, the joint distribution of{xt1 , xt2 , ...., xtm} is the same as the joint distribution of{xt1+h, xt2+h, ...., xtm+h} for all integers h ≥ 1.

• The sequence {xt : t = 1, 2, ....} is identically distributed

• The nature of any correlation between adjacent terms is the sameacross all time periods



Covariance stationary process

Definition (Covariance Stationary Process)

A stochastic process {xt : t = 1, 2, ....} with finite second moment[E (xt) <∞] is covariance stationary if (i) E (xt) is constant; (ii) Var(xt) isconstant; (iii) for any t, h ≥ 1,Cov(xt , xt+h) depends only on h and noton t.

The focus is on the first two moments of a stochastic process

• The mean and variance of the process are constant across time

• The covariance between xt and xt+h depends only on the distancebetween the two terms, h, and not on the location of the initial timeperiod, t

• Thus, correlation between xt and xt+h also depends only on h



If a stationary process has a finite second moment, then it must becovariance stationary, but the converse is not true



Why do we need to assume stationarity?

• If we want to understand the relationship between two or morevariables using regression analysis, we need to assume some sort ofstability over time

• In multiple regression models for time series data, we usually assumea certain form of stationarity in that the βj does not change over time


Further issues of using OLS with TS data Weakly dependent time series

Weakly dependent time series

• Stationarity has to do with the joint distributions of a process as itmoves through time

• A very different concept is that of weak dependence, which placesrestrictions on how strongly related the random variables xt and xt+h

can be as the time distance between them (h) gets large



Weakly dependent time series

• A covariance stationary time series is weakly dependent if thecorrelation between xt and xt+h goes to zero “sufficiently quickly” ash→∞

• As the variables get farther apart in time, the correlation betweenthem becomes smaller and smaller



Weakly dependent time seriesWhy is weak dependence important for regression analysis?

Because it replaces the assumption of random sampling in implying thatthe law of large numbers (LLN) and the central limit theorem (CLT) hold.

• For LLN and CLT to hold, we need that the individual observationsmust not be too strongly related to each other

• The most well-known CLT for time series data requires stationarityand some form of weak dependence

• stationary, weakly dependent time series are ideal for use in multipleregression analysis



Example (Moving average process)

One example of a weakly dependent sequence is:

xt = et + α1et−1 , t = 1, 2...

where {et : t = 0, 1, . . .} is an i.i.d. sequence with zero mean and varianceσ2e .

The process {xt} is called a moving average process of order one

[MA(1)]: xt is a weighted average of et and et−1



The MA(1) process weakly dependent because adjacent terms in thesequence are correlated

xt+1 = et+1 + α1et

Cov(xt , xt+1) = α1Var(et) = α1σ2e

Var(xt) = (1 + α21)σ2

e

Corr(xt , xt+1) =α1

(1 + α21)

However, when we look at variables in the sequence that are two or moretime periods apart, these are uncorrelated because they are independent

Example

xt+2 = et+2 + α1et+1 is independent of xt because {et} is independentacross t



Example (Autoregressive process)

Another example is an autoregressive process of order one [AR(1)]

yt = ρ1yt−1 + et t = 1, 2, ....

The starting point in the sequence is y0 (at t = 0), and {et : t = 1, 2, . . .}is an i.i.d. sequence with zero mean and variance σ2

e .

The crucial assumption for weak dependence of an AR(1) process is thestability condition |ρ1| < 1

This is called a [stable AR(1) process]



• Assuming that the AR(1) process is covariance stationary, we knowthat

E (yt) = E (yt−1)

• If we have that ρ1 6= 1 the above can only happen if

E (yt) = 0

• Taking the variance and using the fact that et and yt are independent(and thus uncorrelated) we have that

Var(yt) = ρ21Var(yt−1) + Var(et)

• Thus, under covariance stationarity we must have

σ2y = ρ2

1σ2y + σ2

e



• Since ρ21 < 1 by the stability condition, we can solve for σ2

y

σ2y =

σ2e

1− ρ21

• To find the covariance between yt and yt+h for h ≥ 1 we do repeatedsubstitution

yt+h = ρ1yt+h−1 + et+h

yt+h = ρ1 (ρ1yt+h−2 + et+h−1) + et+h

yt+h = ρ21yt+h−2 + ρ1et+h−1 + et+h

...

yt+h = ρh1yt + ρh−11 et+1 + ...+ ρ1et+h−1 + et+h



Since E (yt) = 0 for all t, we can multiply this last equation by yt and takeexpectations to obtain Cov(yt , yt+h). Using the fact that et+j isuncorrelated with yt for all h ≥ 1 gives

Cov(yt , yt+1) = E (ytyt+1)

Cov(yt , yt+1) = ρh1E (y2t ) + ρh−1

1 E (ytet+1) + ...+ E (ytet+h)

Cov(yt , yt+1) = ρh1E (y2t )

Cov(yt , yt+1) = ρh1σ2y



• Since σy is the standard deviation of both yt and yt+1, we can easilyfind the correlation betweenyt and yt+1 for any h ≥ 1

Corr(yt , yt+1) =Cov(yt , yt+1)

σyσy= ρh1

• The above expression shows that, while yt and yt+1 are correlated forany h ≥ 1, this correlation gets very small for large h


Asymptotic properties of OLS

Assumption TS1’

Assumption TS1’ [Linearity and weak dependence]: This is the sameas TS1, except we must also assume that {(xt ,yt) : t = 1, 2..} is weaklydependent.

• In other words, the law of large numbers and the central limittheorem can be applied to sample averages

• The xtj can contain lagged dependent and independent variables,provided the weak dependence assumption is met.



Assumption TS2’

Assumption TS2’ [Zero conditional mean]: For each t, E (ut |xt) = 0

This is much weaker than Assumption TS2 because it puts no restrictionson how ut is related to the explanatory variables in other time periods



Assumption TS3’

Assumption TS3’ [No perfect collinearity] Same as Assumption TS3



Theorem 11.1 - Consistency of OLS

Theorem (11.1 Consistency of OLS)

Under TS1’, TS2’ and TS3’, the OLS estimators are consistent:

p lim βj = βj , j = 0, 1, ....., k

Theorem (10.1 Unbiasedness of OLS)

Under assumptions T1, T2 and T3, the OLS estimators are unbiasedconditional on X, and therefore unconditionally as well

E (βj) = βj , j = 0, 1, ....k



Differences between Theorems 10.1 and11.1

• In Theorem 11.1, we conclude that the OLS estimators are consistent,but not necessarily unbiased

• In Theorem 11.1, we have weakened the sense in which theexplanatory variables must be exogenous, but weak dependence isrequired in the underlying time series

NB: weak dependence is also crucial in obtaining approximatedistributional results



Why is it important to relax the strictexogeneity assumption?

1. Strict exogeneity is a serious restriction because it rules out all kindsof dynamic relationships between explanatory variables and the error term

2. In particular, it rules out feedback from the dependent variable onfuture values of the explanatory variables (which is very common ineconomic contexts)

3. Strict exogeneity precludes the use of lagged dependent variables asregressors



Why do lagged dependent variables violatestrict exogeneity?

yt = β0 + β1yt−1 + ut

• Contemporanous exogeneity:

E (ut |yt−1) = 0

• Strict exogeneity:E (ut |y0,y1,, ....yn−1) = 0

Strict exogeneity would imply that the error term is uncorrelated with allyt , t = 1, ..., n − 1

• This is the simplest possible regression model with a laggeddependent variable



This leads to a contradiction because..

Cov(yt , ut) = β1Cov(yt−1, ut) + Var(ut) > 0

OLS estimation in the presence of lagged dependent variables

• Under contemporaneous exogeneity, OLS is consistent but biased



Assumption TS4’

Assumption TS4’ (Homoscedasticity)

Var(ut |xt) = Var(ut) = σ2

The errors are contemporaneously homoscedastic



Assumption TS5’

Assumption TS5’ (No serial correlation)

Corr(ut , us |xt , xs) = 0 t 6= s

Conditional on the explanatory variables, in periods t and s, the errors areuncorrelated



Theorem 11.2

Theorem (11.2 Asymptotic normality of OLS)

Under assumptions TS1’ - TS5’, the OLS estimators are asymptoticallynormally distributed. Further, the usual OLS standard errors, t-statisticsand F-statistics are asymptotically valid



Using trend-stationary series in regressionanalysis

• Time series with deterministic time trends are nonstationary

• If they are stationary around the trend and in addition weaklydependent, they are called trend-stationary processes

• Trend-stationary processes also satisfy assumption TS1’



Using highly persistent time series inregression analysis

• Unfortunately many economic time series violate weak dependencebecause they are highly persistent (= strongly dependent)

• In this case OLS methods are generally invalid (unless the CLM hold)

• In some cases transformations to weak dependence are possible



Example (Efficient Markets Hypothesis (EMH))

The EMH in a strict form states that information observable to the marketprior to week t should not help to predict the return during week t.A simplification assumes in addition that only past returns are consideredas relevant information to predict the return in week t. This implies that

E (rt |rt−1, rt−2..) = E (rt)

A simple way to test the EMH is to specify an AR(1) model. Under theEMH assumption,TS3’ holds so that an OLS regression can be used totest whether this week‘s returns depend on last week‘s

rt = 0.180 + 0.059rt−1

= (.081) (.038)

n = 689, R2 = 0.035, R2 = 0.002

• There is no evidence against the EMH. Including more lagged returnsyields similar results


Random Walk

Random walk

The random walk is called random walk because it wanders from theprevious position yt−1 by an i.i.d. random amount et

yt = yt−1 + et

⇒ yt = (yt−2 + et−1) + et = .. = et + et−1 + ..+ e1 + y0

• The value today is the accumulation of all past shocks plus an initialvalue

• This is the reason why the random walk is highly persistent: The effectof a shock will be contained in the series forever


Random Walk

Figure : Random walk realisations


Random Walk

Figure : Three-month T-bill rate as a possible example for a random walk


Random Walk

Three-month T-bill rate as a possible example for a random walk

• A random walk is a special case of a unit root process

• Unit root processes are defined as the random walk but et may be anarbitrary weakly dependent process.

• From an economic point of view it is important to know whether atime series is highly persistent. In highly persistent time series, shocksor policy changes have lasting/permanent effects, in weaklydependent processes their effects are transitory.


Random Walk

Random walk with drift

• In addition to the usual random walk mechanism, there is adeterministic increase/decrease (= drift) in each period

yt = α0 + yt−1 + et

⇒ yt = α0t + et + et−1 + ..+ e1 + y0

• This leads to a linear time trend around which the series follows itsrandom walk behaviour. As there is no clear direction in which therandom walk develops, it may also wander away from the trend


Random Walk

Random walk with drift

• Otherwise, the random walk with drift has similar properties as therandom walk without drift

E (yt) = α0t + E (yo)

Var (yt) = σ2y t

Corr(yt , yt+h) =

√t

t + h

• Random walks with drift are not covariance stationary and not weaklydependent


Random Walk

Sample path of a random walk with drift

Figure : Random walk realisations

• Note that the series does not regularly return to the trend line• Random walks with drift may be good models for time series that

have an obvious trend but are not weakly dependent


Random Walk

Transformations on highly persistent timeseries

Order of integration

• Weakly dependent time series are integrated of order zero I(0)

• If a time series has to be differences one time in order to obtain aweakly dependent series, it is called integrated of order one I(1)


Random Walk

Transformations on highly persistent timeseries

Example (I(1) processes)

yt = yt−1 + et ⇒ ∆yt = yt − yt−1 = et

ln (yt) = ln (yt−1) + et ⇒ ∆ ln (yt) = et

• After differencing, the resulting series are weakly dependent (becauseet is weakly dependent)

• Differencing is often a way to achieve weak dependence


Random Walk

Deciding whether a time series is I(1)

• There are statistical tests for testing whether a time series is I(1)(unit root tests)

• Alternatively, look at the sample first order autocorrelation

ρ1 = corr(yt , yt−1)

• Measures how strongly adjacent times series observations are relatedto each other


Random Walk

Deciding whether a time series is I(1)

• If the sample first order autocorrelation is close to one, this suggeststhat the time series may be highly persistent (contains a unit root)

• Alternatively, the series may have a deterministic trend

• Both unit root and trend may be eliminated by differencing


Random Walk

Example (Fertility equation)

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

• This equation could be estimated by OLS if the CLM assumptionshold. These may be questionable, so that one would have to resort tolarge sample analysis

• For large sample analysis, the fertility series and the series of thepersonal tax exemption have to be stationary and weakly dependent

• This is questionable because the two series are highly persistent

ρgfr = .977

ρgpe = .964


Random Walk

Example (Fertility equation)

It is therefore better to estimate the equation in first differences. Thismakes sense because if the equation holds in levels, it also has to hold infirst differences:

∆gfrt = −0.964− 0.036∆pet − 0.014∆pet-1 + 0.110∆pet-2

(0.468) (0.027) (0.028)

n = 69,R2 = 0.233,R2

= 0.197


Random Walk

Example (Wages and productivity)

• Include trend because both series display clear trends

• The elasticity of hourly wage with respect to output per hour(=productivity) seems implausibly large

ln(hrware) = −5.33 + 1.64 ln(outphr)− 0.018t

(0.37) (0.09) (0.002)

n = 41,R2 = 0.971,R2

= 0.970


Random Walk

Example (Wages and productivity)

• It turns out that even after detrending, both series display sampleautocorrelations close to one so that estimating the equation in firstdifferences seems more adequate:

∆ ln(hrware) = −0.0036 + 0.809∆ ln(outphr)

(0.0042) (0.173)

n = 40,R2 = 0.364,R2

= 0.348

This estimate of the elasticity of hourly wage with respect toproductivity makes much more sense


Random Walk

Dynamically complete models

A model is said to be dynamically complete if enough lagged variableshave been included as explanatory variables so that further lags do nothelp to explain the dependent variable

E (yt |xt , yt−1, xt−1, yt−2, xt−2..) = E (yt |xt)

Dynamic completeness implies absence of serial correlation

• If further lags actually belong in the regression, their omission willcause serial correlation (if the variables are serially correlated)

One can easily test for dynamic completeness

• If lags cannot be excluded, this suggests there is serial correlation


Random Walk

Sequential exogeneity

A set of explanatory variables is said to be sequentially exogenous if”enough” lagged explanatory variables have been included:

E (ut |xt , xt−1, ..) = E (ut) = 0

• Sequential exogeneity is weaker than strict exogeneity

• Sequential exogeneity is equivalent to dynamic completeness if theexplanatory variables contain a lagged dependent variable


Random Walk

Sequential exogeneity

Should all regression models be dynamically complete?Not necessarily: If sequential exogeneity holds, causal effects will becorrectly estimated; absence of serial correlation is not crucial


Random Walk

Summary

1. OLS can be justified using asymptotic analysis, provided certainconditions are met

2. Weak dependence is necessary for applying the standard largesample results, particularly the central limit theorem

3. Processes with deterministic trends that are weakly dependent canbe used directly in regression analysis, provided time trends are included inthe model


Random Walk

Summary

4. When the time series are highly persistent (they have unit roots), wemust exercise extreme caution in using them directly in regression models

5. An alternative to using the levels is to use the first differences of thevariables (but this changes the nature of the model !!)

6. When models have complete dynamics errors will be seriallyuncorrelated

7. In static and distributed lag models, the dynamically completeassumption is often false, which generally means the errors will be serially


ec3090 - econometrics - time series econometrics · 2015. 12. 4. · time series econometrics agust...

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