TIME SERIES REGRESSION

TS1-5

TS1: linear specifications

TS2: no perfect collinearity

TS3: strict exogeneity (an extension of contemporaneous exogeneity)

TS4: homoscedasticity of U – var(ut) = σ2

TS5: no serial correlation (between the Us)

Gauss-markov : BLUE

Importance for unbiasedness and consistency

DETRENDING

Linear trend model

Partialling out method

STATIONARITY AND WEAK DEPENDENCE

Used in LLN and CLT (so we can assume normality and construct Cis and tests)

Stationarity:

- covariance stationary (weaker form) E(X) and var(x) is constant, and Cov (xt, xt+h) is dependent only on h, not t

Weak dependence:

- xt and xt+h almost independent as h infinity

EDITING THE TS

TS2 holds as per no perfect collinearity

TS1 becomes edited for stationarity and weak dependence

TS3 of strict exogeneity becomes contemporaneous exogeneity so that we can have lagged regressors.

Under these, we can have consistent but biased OLS estimators

E.g. AS(1) where Yt = B0 + B1 Yt-1 + Ut

And E(Ut|Yt-1, Yt-2,…) = 0 contemporaneous exo

And we hold B1 < 1

However, if we have TS4’ and TS5’ which are var(Ut|Xt) = 0 and E(UtUs|XsXt) = 0 then under Ts1-5 the OLS estimators are asymptotically normal.

T stats, F stats, and LM stats are valid.

EFFICIENT MARKETS HYPOTHESIS (AR) The coefficient is not statistically significant

Hence, there is no arbitrage in the markets

HIGHLY PERSISTENT TIME SERIES No weak dependence (correlation does not decrease)

So we transform the data

E.g. Random walk !

Yt = yt-1 + ut

Not covariance stationary

But weakly dependent

one generalization of RW is the unit root process which has consistent coefficients but not normal distribution (B is biased towards 0)

another is the RW with drift = introducing α, which introduces a trend in the E(y)

So we transform by

- first difference a random walk

- first difference will also remove time trend from linear trend model

NEXT LECTURE…

SERIAL CORRELATION AND HETEROSKEDASTICITY TS 4 AND 5 Unbiasedness requires TS 1- 3

Consistency requires TS1’ – 3’

These are regardless of serial correlation

We have TS 1 – 5’ to have OLS BLUE

But with serial correlation, violation of TS5’ no longer BLUE

VARIANCE OF OLS W S.C.

With S.C., corr(U1, U2) is positive.

With lag dependent variables, there is no more strict exogeneity since the error term will depend on the Xs