1 Identifying ARIMA Models What you need to know

2 Autoregressive of the second order X(t) = b 1 x(t-1) + b 2 x(t-2) + wn(t) b 2 is the partial regression coefficient measuring the effect of x(t-2) on x(t) holding x(t-1) constant Since x(t) is regressed on itself lagged, b 2 can also be interpreted as a partial autoregression coefficient of x(t) regressed on itself lagged twice.

3 continued In one more step b 2 can be defined as the partial autocorrelation coefficient at lag 2, b 2 = pacf(2) Solving the yule-Walker equations: b 2 = {acf(2) – [acf(1)] 2 }/[1 – [acf(1)] 2 We know that if the process is autoregressive of the first order, then acf(2) = [acf(1)] 2 and so b 2 = 0

4 So now we are back to autoregressive of the first oder x(t) = b x(t-1) + wn(t) There is only one regression coefficient, b, so acf(1) = pacf(1) = b

5 In summary The partial autocorrelation function, pacf(u) indicates the order of the autregressive process. If only pacf(1) is significantly different from zero, then the autoregressive process is of order one. If the pacf(2) is significantly different from zero, then the autoregressive process is of order two, and so on. Thus we use the partial autocorrelation function to specify the order of the autoregressive process to be estimated

6 The autocorrelation function The autocorrelation function, acf(u) is used to determine the order of the moving average process If acf(1) is significantly different from zero and there are no other significant autocorrelations, then we specify a first order MA process to be estimated

7 Cont. If there is a significant autocorrelation at lag two and none at higher lags, then we specify a second order moving average process

8 Moving Average Process X(t) = wn(t) + a 1 wn(t-1) + a 2 wn(t-2) + a 3 wn(t-3) Taking expectations the mean function is zero, Ex(t) = m(t) = o Multiplying by x(t-1) and taking expectations, E[x(t)x(t-1)] = EX(t) = wn(t) + a 1 wn(t-1) + a 2 wn(t-2) + a 3 wn(t-3) X(t-1) = wn(t-1) + a 1 wn(t-2) + a 2 wn(t-3) + a 3 wn(t- 4), γ x,x (1) = [a 1 + a 1 a 2 + a 2 a 3 ] σ 2

9 Continuing The autocovariance at lag 2, γ x,x (2) = E x(t) x(t-2) EX(t) = wn(t) + a 1 wn(t-1) + a 2 wn(t-2) + a 3 wn(t-3) X(t-2) = wn(t-2) + a 1 wn(t-3) + a 2 wn(t-4) + a 3 wn(t-5), γ x,x (2) = [a 2 + a 1 a 3 ] σ 2 The autocovariance at lag 3, γ x,x (3) = E x(t) x(t-4) EX(t) = wn(t) + a 1 wn(t-1) + a 2 wn(t-2) + a 3 wn(t-3) X(t-3) = wn(t-3) + a 1 wn(t-4) + a 2 wn(t-5) + a 3 wn(t-6), γ x,x (3) = [a 3 ] σ 2 The autocovariance at lag 4 is zero, E x(t)x(t-4) = 0, so the autocovariance function determines the order of the MA process

10 Specifying ARMA Processes x(t) = A(z)/B(z) The autocovariance function divided by the variance, i.e. the autocorrelation function, acf(u), indicates the order of A(z) and the partial autocorrelation function, pacf(u) indicates the order of B(z) In Eviews specify x(t) c ar(1) ar(2) ….ar(u) for a u th order B(z) and include ma(1) ma(2) ….ma(u) for a u th order A(z), i.e. X(t) c ar(1) ar(2) …ar(u) ma(1) ma(2) …ma(u)

11 Summary of Identification Spreadsheet Trace: Is it stationary? Histogram: is it normal? Correlogram: order of A(z) and B(z) Unit root test: is it stationary?1111 Specification estimation

12 ARMA Processes Identification Specification and Estimation Validation –Significance of estimated parameters and DW –Actual, fitted and residual –Residual tests Correlogram: are they orthogonal? Also the Breusch-Godfrey test for serial correlation Histogram; are they normal? Forecasting

13 Example: Capacity utilization mfg.

14Spreadsheet

15Histogram

16 Correlogram

17 Unit root test

18 Pre-Whiten Gen dmcumfn =mcumfn – mcumfn(-1)

19Spreadsheet

20 Trace

21histogram

22Correlogram

23 Unit root test

24 Specification Dmcumfn c ar(1) ar(2)

25 Estimation

26Validation

27 Correlogram of the residuals

28 Breusch-Godfrey Serial correlation test

29 Re-Specify

30 Estimation

31 Validation

32 Correlogram of the Residuals

33 Breusch-Godfrey Serial correlation test

34 Histogram of the residuals

35 Forecasting: Procs. Workfile range

36 Forecasting: Equation window.forecast

37 Forecasting

38 Forecasting: Quick, show

39Forecasting

40 Forecasting: show, view, graph-line

41 Reintegration

42 Forecasting mcumfn

43 Forecast mcumfn, quick, show

44 Forecasting mcumfn

45 What can we learn from this forecast? If, in the next nine months, mcumfn grows beyond the upper bound, this is new information indicating a rebound in manufacturing If, in the next nine months, mcumfn stays within the upper and lower bounds, then this means the recovery remains sluggish If mcumfn goes below the lower bound, run for the hills!