Time Series

• {Yt} is weakly stationary if its mean, variance and autocovariances do not depend on t
• {Xt} and {Yt} are jointly weakly stationary if each series is weakly stationary and cov(Yt, Xt-h) does not depend on t for all h
• the population autocorrelation function measures the persistence of a time series and is consistently estimated by the sample autocorrelation function
• stationary processes are commonly only weakly persistent
• strong persistence (weakly decaying autocorrelation) is often suggestive of non-stationarity
• {Yt} is strongly stationary if the distribution of every subsequence has the same distribution
• trends are usually dealt with by transforming the data to obtain a stationary process
• To detrend a dataset we might have to take logarithms before differencing, particularly for series where exponential growth/decay makes sense
• the AR(1) model is intended as a descriptive model, not a causal one
• the mean squared forecast error (MSFE) minimising forecast is the expectation of Yt+1 conditional on all past values of Yt
• forecast errors generally have two components: estimation error and the unforecastable component of the model
• advantages to choosing a larger p: more flexible model, potentially better description of the dynamics of Yt, more parameters over which to optimise forecast rule, less 'bias' in the approximation of the optimal forecast
• disadvantages of larger p: more parameters to estimate using the same amount of data, more 'variance' in the estimation of the model parameters
• an AR(p) model has m=p+1 parameters
• BIC penalises larger models more than AIC does
• ARDL(p,q) model has m=p+q+1 parameters
• we say Xt Granger causes Yt if lags of Xt improve forecasts of Yt made on the basis of its own lags (reduce the optimal forecasts MSFE)
• a break is an abrupt change in model parameters on (or very near) a particular date; modelled using breakpoint dummies
• breaks are a leading cause of forecast failure, particularly those near the end of a sample
• if delta Yt follows an AR(p-1) model, Yt follows an AR(p) model with coefficients summing to 1
• deterministic trends generate linear growth
• stochastic trends generate and are synonymous with 'random wandering' behaviour
• unit root AR processes can decompose into the sum of a deterministic trend, a stochastic trend, and an initial value
• unit root processes provide biased OLS estimates, and CLTs do not apply - inference is non-standard
• Dickey-Fuller test is used to test for unit roots
• ADF test may have misleading conclusions when applied to a series with a linear trend
• order of integration is the smallest number of differences required to make a sequence stationary
• spurious regression is the systematic tendency to find statistically significant regression relationships between unrelated I(1) series
• Xt and Yt are cointegrated if there exists a cointegrating coefficient such that Yt-thetaXt ~ I(0)