QE 7-8: Time series

Question 1

Is time series analysis (usually) causal or descriptive? Why?

Answer 1

Descriptive. We only observe one past history, so can rarely make causal inferences.

Question 2

What is strong stationarity?

Answer 2

Joint distributions are time-invariant.

Question 3

What is weak stationarity?

Answer 3

Mean, variance and autocovariances do not depend on t.

Question 4

What are some common sources of non-stationarity?

Answer 4

• Deterministic trends
• Random wandering behaviour - stochastic ctrends
• Structural breaks

Question 5

What are some common techniques to generate stationary series from nonstationary ones?

Answer 5

Can analyse subsamples of data pre- and post-break.

With trends, we can difference the data, or possibly ‘detrend’ after fitting a regression line. Taking logs or log differences can stabilise the variance.

Question 6

What is an AR(1) model? What exactly is the shock sequence it involves?

Answer 6

In an AR(1) model, Yt = b0 + b1Y_{t-1} + ut. u is white noise.

Question 7

Does it matter whether or not the causal model that actually generates Yt is an AR model, for purposes of forecasting?

Answer 7

No, this is just a descriptive model of how Yt is correlated with its lags.

Question 8

What conditions on the model parameters are sufficient for AR(1) being weakly stationary?

Answer 8

b1 is in (-1,1) is sufficient for asymptotic weak stationarity; the initial value Y0 must also have the correct mean and variance for weak stationarity in finite samples.

Question 9

What is asymptotic weak stationarity?

Answer 9

A process is asymptotically weakly stationary if its expectation and variance approach constants in the limit.

Question 10

How can we find optimal 2-step ahead forecasts?

Answer 10

Find the optimal 1-step ahead forecast, and recursively use this in the 2-step ahead forecast.

Question 11

What is the difference between estimation error and shock error?

Answer 11

Estimation error is the error introduced from estimating the optimal forecast. Shock error is the unforecastable shock in the model.

Question 12

What is a necessary requirement on the beta_i for an AR(p) model to be stationary?

Answer 12

Their sum is less than 1.

Question 13

Why does ô2_u underestimate the MSFE?

Answer 13

ô2_u is an estimate of in-sample errors, whereas the MSFE is a measure of errors made when predicting out-of-sample values.

Question 14

What is psuedo out-of-sample forecasting? How can it help us estimate the MSFE?

Answer 14

Pseudo out-of-sample forecasting takes some date s<T and runs the forecasting procedure only on data up to s. Comparing the forecast to the actual data provides an estimate of the MSFE.

Question 15

What is the bias-variance tradeoff?

Answer 15

When selecting a value of p to use in an AR(p) model, we face the following tradeoff:

(i) Larger values of p allow for more flexible modelling, more free parameters, and therefore reduced bias.

(ii) Larger values of p require more parameters to be estimated with a finite amount of data, increasing the variability of those estimates.

Question 16

What is stepwise testing? How can it help us choose a lag order?

Answer 16

We can start with an AR(p_max) model, and then perform a t-test of H_0: b_p =0. If we accept H0, reduce p to p-1 and return to step 1. We continue doing this until we reject the null hypothesis.

Question 17

What are information criteria? Is the aim to maximise or minimise them?

Answer 17

The Akaike and Bayesian information criteria are formulae to evaluate different predictive models. We aim to minimise them, in order to optimise the bias-variance tradeoff.

Question 18

Should we consider models with uneven lags?

Answer 18

This is possible, but increases the number of models to choose from. We increase the probability of choosing a bad model by chance. We should only do this if a priori sensible, eg including quarterly data.

Question 19

What is an ADL(p,q) model?

Answer 19

An autoregressive distributed lag model of order (p,q) for Y_t is a linear function of p lags of Y_t and q lags of a different series X_t.

Question 20

What is Granger causality?

Answer 20

{Xt} does not Granger cause {Yt} if lags of {Xt} carry no useful information about {Yt} in the sense that the optimal forecast of Yt is not improved by Xt. Xt does Granger-cause Yt if the preceding is false.

Question 21

How can a Chow breakpoint test be carried out?

Answer 21

Add a breakpoint dummy variable to the regression, and interact it with all rhs variables. F-test that all these terms equal zero.

Question 22

What is the QLR test? How is it related to the Chow test?

Answer 22

The QLR test finds the Chow statistic for many possible break points and takes the maximum of these.

Question 23

If ∆Y_t is AR(p-1), what process does Y_t follow? What type of process is this?

Answer 23

Yt is AR(p) with ∑bi = 1. This is a unit root process.

Question 24

What happens if we try to estimate the parameters of an AR(p) process with a unit root? Are the estimates consistent? Are they unbiased?

Answer 24

The estimates are consistent, but biased and not asymptotically normal.

Question 25

Why might we want to test for unit roots?

Answer 25

• We may want to find out if standard inferences will be valid on our estimated coefficients.
• We may want to test whether a series is truly non-stationary or simply highly persistent.

Question 26

What is the constant-and-trend DF test? Why might we use it?

Answer 26

We may also want to test against a trend-stationary process, rather than a stationary process.

Question 27

How do we choose the lag-order determination in an augmented DF test?

Answer 27

Stepwise testing, or the information criteria.

Question 28

What is an ‘order of integration’?

Answer 28

The order of integration of a process Yt is the smallest integer d such that ∆^dYt is stationary; denoted Yt ~ I(d).

Question 29

How can the order of integration of a time series be determined?

Answer 29

Test for a unit root. If rejected, Yt ~ I(0); if not, difference continue testing stepwise.

Question 30

Can we regress Y_t on X_t and perform standard inferences if both are I(1)?

Answer 30

No, this type of regression will often lead to spurious results. The t-statistic associated with b1 on a regression between independent I(1) variables Yt and Xt diverges in magnitude as the sample size grows; we will tend to find statistically significant relationships between I(1) series.

Question 31

Define what it means for X_t and Y_t to be cointegrated.

Answer 31

Xt and Yt are said to be cointegrated if there is a constant c such that Yt - cXt ~ I(0).

Question 32

How do we test for cointegration if we know the cointegrating coefficient?

Answer 32

Compute Yt - cXt and conduct an ADF test for a unit root. The null hypothesis corresponds to no cointegration.