FAME Time Series Econometrics Daniel V. Gordon Department of Economics University of Calgary

Time Series Econometrics is difficult Distribution assumption violated Think what we are asking about the distribution: Overtime nothing changes in the mean or variance No learning takes place, no changes in technology, no changes!! Of course, when you think of it things are changing all the time, So what does this mean for the distribution?

Lets start with a single series, say We want to forecast based on assumption that series generated by stochastic or random process Economic behaviour contains dynamic components, shocks in period t 0 can impact in t 0 but also t 1, t 2, etc. With time series data data may have autocorrelated error structure, e t related to e t-1, e t-2, … e t-j Think of related to past values of and current and past values of e t Called ARMA modelling, many extensions to ARIMA or ARMAX models

In TS we characterize a stochastic process (SP) as being stable or stationary in probability; the distribution does not change overtime Stationary Violation of any one of these conditions your distribution is changing over time. Lets give an example of a non-stationary series: Random Wald

Say we are in the middle of the street and we want to get to one side or other, describe the walk by a series by a random variable say can take a value of 2 steps to the left (-2) or 2 steps to the right (+2) Make the process random So mean of = 0 and variance = 4 And certainly independent Lets work through the SP of street walk or the position reached relative to the stating point

The question, is this process stationary? Variance a linear function of time, so not stationary Non-stationary series are very difficult to work with because statistical properties are changing over time, Worse yet results look good but are useless

Example of the problem, let be a series of integers And a series of integers squared Say n=30 Each variable has a deterministic trend, diverges over time. Of course spurious relationship

Trends in economic data dominate in the regression and we lose economic significance. A useful characterization of a SP is the autocorrelation function A measure of correlation A stationary process will decay quickly to zero, so a nice visual procedure for summaries SP Individual can be tested against zero by the Bartlett test where SE of is WE use a Q-test for a joint test that all = zero, No correlation Stata command ‘corrgram’ provides autocorrelation and more good stuff

Must be careful with trends in time series data, one way of accounting for trends is first-differencing Random Walk and know non-stationary first differences stationary If seasonal trends in the SP take seasonal differences Or There are a number of tests for stationary SP we will describe the Dickey- Fuller test and the KSPP test. For DF test Null is non-stationary and Alternative is Stationary in first differences. (help dfuller) For KPSS test Null is stationary and Alternative is Non-stationary (help KPSS)

DF test is that = 0. If < 1 stationary If you reject then SP first-difference stationary Can add in time trends and correction for autocorrelation KPSS test is that or mean is constant Structural breaks can make a stationary series look non-stationary One test zandrews test will allow test for one structural break See,

Need some procedure for setting the length of lags. A number to choose from Say you have j different models and want to choose, assume correct model is one of the choices Start with squared sum of errors Could use this min variance or max R 2 same thing Lots of transformations of this rule Amemiya, AIC, BIC, Both AIC and BIC based on likelihood function Select model that minimizes values One test for functional form, based on the idea that you have defined the correct variables but unsure of functional structure

Ramsey Test Say we have a linear specification Worried about missing non linear terms. Easy test run the regression and predict from this calculate Think of these variables as proxies for non linear terms Rerun the regression including proxies and use an F test for a null of no misspecification