2. Time Series Econometrics Flashcards
What is a time series process?
A set of temporally ordered observations on a variable y taken at equally spaced discrete intervals in time
What has to be true for a stochastic process, y, to be covariance stationary?
Each yt has the same mean and variance, and the covariance between yt and yt-1 depends only on the separation, not on t
What does strict stationarity require?
The joint PDF of yt-s, yt-s+1, …, yt is identical to yt-s+k, yt-s+k+1,…, yt+k
What is a disadvantage of the autocovariance function?
It isn’t scale invariant
What is the Autoregressive Moving Average process (ARMA)
A linear function of lags of itself and of Ęt and lags
What are some properties of Ęt?
-E(Ęt)=0
-V(Ęt)= ó^2
-C(Ęt, Ęt-1) =0
Hence Ęt is itself a covariance stationary process called white noise
What do the mean, variance and autocovariance of the MA(q) process depend on?
They don’t depend on t, therefore the process is stationary for any value of theta
When is the AR(p) process stationary?
When |phi| <1 since the variances and autocovariances don’t depend on t when this is the case
What happens to the non-zero autocorrelations in the AR(1) process when S increases?
The autocorrelations decay to zero because |phi|<1
What does the lag operator do?
Lxt = xt-1
It shifts the period back one
How can we use the lag operator?
As a simple way of moving between MA and AR representations of the ARMA process
Which ARMA(p,q) processes can be written as MA(infinity) or AR(infinity)?
Any stationary and invertible ARMA(p,q)
What does white noise refer to?
A process whose autocorrelations are zero at all logs
If an ARMA(2,1) process has identical autocorrelations to the AR(1) process, what can we say about the ARMA(2,1) process?
It is overparameterised
Equation for random walk
yt= yt-1 +Et
What is f(yt | Yt-1)?
The density of yt given that we know everything up to time t-1
What is lnf(Yt) when y1 isn’t fixed?
The prediction error decomposition form of the log likelihood function
What is lnf(Yt) when we fix y1?
The conditional log likelihood function
What is the CSS?
The conditional sum of squares. It squares then sums the deviation of each yt from its conditional mean phi yt-1
How can we make finding the MLE easier?
We can find the simpler conditional log likelihood function by treating y1 as fixed
What trick do we use to find the MLE of a MA(1) process?
We invert it to a AR(infinity) representation since we can’t observe the unobserved errors in the MA(1)
What can we say about the order of an underlying stochastic process if rho(s) is truncated after x number of lags?
It is an MA(q) process where q=x
What can we say about the order of an underlying stochastic process if phi(ss) is truncated after x number of lags?
It is an AR(p) process where p=x
What can we say about the order of an underlying stochastic process if rho(s) and phi(ss) both did out slowly?
It is an ARMA process but we don’t know p and q, we should initially start with ARMA(1,1) and go from there
What are the residuals for a pure AR model?
They are simply the usual OLS residuals
If we have fitted the correct model what will our Êt (residuals) look like?
They will resemble the true residuals Et and consequently behave like white noise
If an AR(1) model is fitted to data generated by an MA(1) process, how will the modelled residuals Êt behave?
They will behave like an MA(2) process
What does the residual autocorrelation function do?
We can use it to indicate the direction of the model misspecification, although it isn’t a perfect measure
What can we conclude if the residual autocorrelations are insignificant?
That there is no model misspecification
What can we conclude if an AR(1) model is fitted to data and the residuals have two significant autocorrelations?
This would indicate that an MA(1) component has been omitted
What can we conclude if an AR(1) model is fitted to data and the residuals have significant autocorrelations that die away slowly?
Could indicate that a second AR component has been omitted
Why isn’t the residual autocorrelation function perfect?
Because rho “tilda” doesn’t share quite the same distributional properties as rho “hat”
Drawbacks of the Portmanteau statistic
r>p+q needs to be selected and different choices for r can yield different inferences about the presence of model misspecification.
It obscures possibly valuable info about the direction of misspecification contained in the individual rho tilda
When should info criteria be used?
To choose between competing models that appear to satisfy the null hypothesis of no model misspecification
What are the two more common info criteria?
Akaike info criterion (AIC)
Schwartz- Bayes info criterion (SBIC)
How do we judge AIC and SBIC?
The smallest value is best
What is choosing the model with the fewest parameters called?
Principle of parsimony
Which info criteria imposes a harsher penalty function?
SBIC
Why are ARMA models inherently difficult to estimate and when is this particularly the case?
They are hard to hard to estimate because of non linearity associated with ML estimation. This is particularly the case if sample size T is small or the MA order is high
In the case of forecasting what is S?
The forecast horizon
What does ŷT+s | YT denote? (T and S are subscript)
Denotes the predictor of YT+s based on YT (ie uses info only up to time T
In forecasts what is the prediction error given by?
YT+s - ŷT+s | YT
What is the general optimal, (as in minimum forecast MSE) forecast of yT+s?
The conditional mean of yT+s conditional on time T. This has a forecast MSE equal to V(yT+s |T)
In general what is the optimal predictor of yT+s|T and the MSE(ŷT+s|T) for an AR model?
ŷT+s|T= phi ^(s) yT
MSE(ŷT+s|T) = ó^2 x (1-phi^(2s))/ (1-phi^2)
In general what is the optimal predictor of yT+s|T and the MSE(ŷT+s|T) for an MA model?
ŷT+s|T = -theta x ET when s=1,
and 0 when s>1
MSE(ŷT+s|T) = ó^2 when s=1,
And ó^2 x (1+theta^2) when s>1
In general what is the optimal predictor of yT+s|T and the MSE(ŷT+s|T) for an ARMA model?
ŷT+s|T= phi x yT - theta x ET when s=1,
And phi^(s-1) x ŷT+s|T when s>1
MSE(ŷT+s|T)= ó^2
What are optimal forecast functions similar to and why?
They are similar to ACF’s. The forecast function concerns the relationship between yt+s and yt whilst the ACF considers the relationship between yt and yt-s which for a stationary process is the same thing
As s goes to infinity what happens to our prediction ŷT+s|T?
It goes to 0=E(yt)
As s goes to infinity what happens to our prediction MSE(ŷT+s|T)?
MSE(ŷT+s|T) goes to V(yt)
When do we need to aware of spurious regressions?
Whenever the variables involved in a fitted regression model might be suspected of being random walks, especially when T is large
How is the autocorrelation function an improvement on the autocovariance function?
The autocorrelation function is scale invariant
What is the equation for the autocorrelation function?
¥s/¥o
In words, the covariance of yt and yt-s divided by the variance of yt
