QE 7-8: Time series Flashcards
Is time series analysis (usually) causal or descriptive? Why?
Descriptive. We only observe one past history, so can rarely make causal inferences.
What is strong stationarity?
Joint distributions are time-invariant.
What is weak stationarity?
Mean, variance and autocovariances do not depend on t.
What are some common sources of non-stationarity?
- Deterministic trends
- Random wandering behaviour - stochastic ctrends
- Structural breaks
What are some common techniques to generate stationary series from nonstationary ones?
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.
What is an AR(1) model? What exactly is the shock sequence it involves?
In an AR(1) model, Yt = b0 + b1Y_{t-1} + ut. u is white noise.
Does it matter whether or not the causal model that actually generates Yt is an AR model, for purposes of forecasting?
No, this is just a descriptive model of how Yt is correlated with its lags.
What conditions on the model parameters are sufficient for AR(1) being weakly stationary?
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.
What is asymptotic weak stationarity?
A process is asymptotically weakly stationary if its expectation and variance approach constants in the limit.
How can we find optimal 2-step ahead forecasts?
Find the optimal 1-step ahead forecast, and recursively use this in the 2-step ahead forecast.
What is the difference between estimation error and shock error?
Estimation error is the error introduced from estimating the optimal forecast. Shock error is the unforecastable shock in the model.
What is a necessary requirement on the beta_i for an AR(p) model to be stationary?
Their sum is less than 1.
Why does ô2_u underestimate the MSFE?
ô2_u is an estimate of in-sample errors, whereas the MSFE is a measure of errors made when predicting out-of-sample values.
What is psuedo out-of-sample forecasting? How can it help us estimate the MSFE?
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.
What is the bias-variance tradeoff?
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.
What is stepwise testing? How can it help us choose a lag order?
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.
What are information criteria? Is the aim to maximise or minimise them?
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.
Should we consider models with uneven lags?
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.
What is an ADL(p,q) model?
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.
What is Granger causality?
{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.
How can a Chow breakpoint test be carried out?
Add a breakpoint dummy variable to the regression, and interact it with all rhs variables. F-test that all these terms equal zero.
What is the QLR test? How is it related to the Chow test?
The QLR test finds the Chow statistic for many possible break points and takes the maximum of these.
If ∆Y_t is AR(p-1), what process does Y_t follow? What type of process is this?
Yt is AR(p) with ∑bi = 1. This is a unit root process.
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?
The estimates are consistent, but biased and not asymptotically normal.
Why might we want to test for unit roots?
- 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.
What is the constant-and-trend DF test? Why might we use it?
We may also want to test against a trend-stationary process, rather than a stationary process.
How do we choose the lag-order determination in an augmented DF test?
Stepwise testing, or the information criteria.
What is an ‘order of integration’?
The order of integration of a process Yt is the smallest integer d such that ∆^dYt is stationary; denoted Yt ~ I(d).
How can the order of integration of a time series be determined?
Test for a unit root. If rejected, Yt ~ I(0); if not, difference continue testing stepwise.
Can we regress Y_t on X_t and perform standard inferences if both are I(1)?
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.
Define what it means for X_t and Y_t to be cointegrated.
Xt and Yt are said to be cointegrated if there is a constant c such that Yt - cXt ~ I(0).
How do we test for cointegration if we know the cointegrating coefficient?
Compute Yt - cXt and conduct an ADF test for a unit root. The null hypothesis corresponds to no cointegration.
