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FAME Time Series Econometrics

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

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FAME Time Series Econometrics

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

  2. 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?

  3. 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 t0 can impact in t0 but also t1, t2, etc. • With time series data data may have autocorrelated error structure, • et related to et-1 , et-2, … et-j • Think of related to past values of and current and past values of et • Called ARMA modelling, many extensions to ARIMA or ARMAX models

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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)

  10. 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, http://www.stata-journal.com/article.html?article=st0080

  11. 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 R2 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

  12. 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

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