Univariate Time series - 2 Methods of Economic Investigation Lecture 19

Last Time  Concepts that are useful Stationarity Ergodicity Ergodic Theorem Autocovariance Generating Function Lag Operators  Time Series Processes AR(p) MA(q)

Today’s Class  Building up to estimation Wold Decomposition Estimating with exogenous, serially correlated errors Testing for Lag Length

Refresher  Stationarity: Some persistence but not too much  Ergodicity: Persistence dies out entirely over some finite period of time  Square Summability (assumption for MA process) with parameter θ such that  Invertibility (assumption for AR process) with parameter φ which has roots λ such that

ARMA process  In general can have a process with both AR and MA components  A general ARMA(p, q) process in our lag function notation this looks like:a(L)x t = b(L)ε t For example, we may have an ARMA(2, 1) x t – (φ 1 x t-1 + φ 2 x t-3 ) = ε t + θ 1 ε t-1 (1 - φ 1 L – φ 2 L 2 ) x t = (1 + θ 1 L) ε t  If the process is invertible then we can rewrite this as: x t =a(L) -1 b(L) ε t

Why Focus on ARMA processes  Define the range of ARMA processes (invertible AR lag polynomials, square summable MA lag polynomials) which can rely on convergence theorems  any time series that is covariance stationary, has a linear ARMA representation.

Information Sets  At time t-n Everything for time t-n and before is known Everything at time t is unknown Information set Ω t-n  Define E t-n (ε t ) = E[ε t | Ω t-n ] Distinct from E[ε t ] because we know previous values of ε’s up until t-n For example, suppose n = 1 and ε t = π ε t-1 +η,  E (ε t ) =0 for all t so it’s a mean zero process  E t-1 (ε t ) =π ε t-1

Recalling the CEF  Define the linear conditional expectation function CEF(a | b) which is the linear project, i.e. the fitted values of a regression of a on b. i.e. a = βb  This is distinct from the general expectations operator in that it is imposing a linear form of the conditional expectation function.

Wold Decomposition Theorem - 1  Formally the Wold Decomposition Theorem says that:  Any mean zero weakly stationary process {x t } can be represented in the form  This comes with some properties for each term…

Wold Decomposition Theorem - 2  Where  ε t ≡ x t − CEF(x t | x t-1, x t-2,...,x 0 ).  Properties of ε t CEF (ε t |x t−1, x t−2,... x 0 )=0, E(ε t x t−j ) = 0, E(ε t ) = 0, E(ε t 2 ) = σ 2 for all t, and E(ε t ε s ) = 0 for all t ≠s  The MA polynomial is invertible  The parameters θ is square summable  {θ j } and {ε s } are unique.  η t is linearly deterministic i.e. η t = CEF(η t |x t−1,....).

A note on the Wold Decomposition  Much of the properties come directly from our assumptions tha the process is weakly stationary  While it says mean zero process, remember we can de-mean our data so most processes can be represented in this format.

Uses of Wold Form  This theorem is extremely useful because it returns time-series processes back to our standard OLS model. Notice that we’ve relaxed some of the conditions for the Gauss-Markov theorem to hold.  the Wold MA(∞) representation is unique. if two time series have the same Wold representation, then they are the same time series This on true only up to second moments in linear forecasting 

Emphasis on Linearity  although CEF(ε t | x t−j ) = 0, can have E(ε t | x t−j ) ≠ 0 with nonlinear projections  If the true x t is not generated by linear combinations of past x t plus a shock, then the Wold shocks ( ε’s) will be different from the true shocks.  The uniqueness result only states that the Wold representation is the unique linear representation where the shocks are linear forecast errors.

Estimating with Serially Correlated Errors  Suppose that we have: Y t = βX t + ε t E[ε t | x t ] = 0, E[ε t 2 | x t ]=σ 2 E[ ε t ε t-k ] = γ k for k≠0 and so define E[ε t ε k ] = σ 2 Γ We could consistently estimate β but our standard errors would be incorrect making it difficult to do inference. Just a heteroskedasticity problem which we have already seen with random effects  Use feasible GLS to estimate weights and then re-estimate OLS to obtain efficient standard errors.

Endogenous Lagged Regressors  May be the case that either the dependent variable or the regressor should enter the estimating equation in lag values too  Suppose we were estimating t Y t = β 0 X t +β 1 X t-1 + … β k X t-k + ε t. We think that these X’s are correlated with Y up to some lag length k We think these X’s are correlated with each other (e.g. some underlying AR process) but we’re not sure how many lags to include

Naive Test  Include lags going very far back r >> k  test the longest lag coefficient β r = 0 and see if that is significant. If not, drop it and keep going.  Problems: Practically, the longer lags you take, the more data you make unusable because it doesn’t have enough time periods to construct the lags. doesn’t allow lag t-6 but exclude lag t-3. The theoretical issue is that we will reject the null 5 percent of the time, even if it’s true (or whatever the significance of the test is).

More sophisticated testing  Can be a bit more sophisticated comparing restricted and unrestricted models define p max as some long lag length greater than the expected relevant lag length In general, we do not test our p max but as before, as p  p max the sample size decreases. Define ε j = Y t = β 0 X t +β 1 X t-1 + … β j X j and let N be the sample size. We therefore could imagine trying to minimize the sum of squared residual:

Cost Functions  Intuition: c(. ) is a penalty for adding additional parameters thus we try to pick the best specification using that cost function to penalize inclusion of extra but irrelevant lags. Akaike (AIC): c(n) = 2  the AIC criterion is not well-founded in theory and will be biased in finite samples  the bias will tend to overstate the true lag length Bayesian: c(n) = log(n)  the BIC will converge to the true p.

Return to Likelihood Ratio Tests  The minimization problem is just likelihood ratio test  To see this, compare lag length j to lag length k. We can write:  Define constant LR test Constant: Declining in N

Next Time  Multivariate Time Series Testing for Unit Roots Cointegration  Returning to Causal Effects Impulse Response Functions Forecasting