▶ Set up the workfile (using menu)

1. Go to File > Open > Foreign Data as Workbook… and select your Excel or CSV file. Follow the wizard and check that the date structure is detected as Monthly.
2. To extend the range for forecasting, go to Proc > Structure/Resize Current Page… and change the End date to 2021m11.
3. Save the workfile with File > Save As….

▶ EViews Command (type in the Command window)

wfopen "C:\data\cpi.xlsx"      ' import the data (change the path)
pagestruct(end=2021m11)         ' extend the range by 12 months for forecasts
smpl 2009m02 2020m11            ' set the sample to the observed data

▶ EViews Steps (menu path)

1. Double-click the series cpi in the workfile window to open it.
2. Click View > Graph….
3. Under Graph type > Basic type, choose Line & Symbol (or Line) and click OK.

▶ EViews Command (type in the Command window)

cpi.line

✎ How to interpret this output

CPI climbs from roughly 212 at the start of the sample to about 260 at the end. This persistent upward movement means the mean is not constant over time, which is the first sign of non-stationarity. There is also a short-lived dip around 2020 that stands out from the otherwise smooth path. No seasonal pattern is visible to the eye, so we proceed with a non-seasonal ARIMA model.

Conclusion of Step 1: the series is trending and probably non-stationary; we confirm this with the correlogram and a formal unit root test.

▶ EViews Steps (menu path)

1. With cpi open, click View > Correlogram….
2. Under Correlogram of, select Level.
3. Set Lags to include to 36 (a reasonable choice for monthly data, i.e. three years) and click OK.

▶ EViews Command (type in the Command window)

cpi.correl(36)

✎ How to interpret this output

EViews shows the autocorrelation function (AC), the partial autocorrelation function (PAC), the Ljung-Box Q-statistic, and its p-value (Prob) for each lag.

• AC: the first autocorrelation is 0.976, and the values decay very slowly: 0.721 at lag 12, 0.435 at lag 24, and still 0.208 at lag 36. A stationary series would show a much faster decay.
• PAC: a single large spike at lag 1 (0.976), after which the partial autocorrelations are close to zero.
• Q-statistic: every Prob is 0.000, so the null of no autocorrelation is rejected at all lags.

Conclusion of Step 2: slow-decaying ACF together with a PACF that cuts off after lag 1 at a value close to one is the classic fingerprint of a non-stationary (unit root) series. We now confirm it formally.

▶ EViews Steps (menu path)

1. Double-clickcpi to open it, click View > Unit Root Tests > Standard Unit Root Test….
2. Test type: Augmented Dickey-Fuller.
3. Test for unit root in: Level.
4. Include in test equation: Trend and intercept (since the series has a trend).
5. Lag length: Automatic selection, Schwarz Information Criterion, and maximum lags 13 (EViews proposes this value automatically for 142 monthly observations).
6. Click OK.

▶ EViews Command (type in the Command window)

cpi.uroot(adf, trend, info=sic, maxlag=13)

✎ How to interpret this output

Compare the ADF t-statistic with the critical values. The statistic (-2.3231) is less negative than even the 10% critical value (-3.1459), and the p-value (0.4184) is far above 0.05. We therefore fail to reject the null hypothesis: CPI has a unit root and is non-stationary in levels. This agrees with the graph and the correlogram.

▶ EViews Steps (menu path)

1. Reopen the unit root dialog: View > Unit Root Tests > Standard Unit Root Test….
2. Change Test for unit root in to 1st difference. Keep the other settings and click OK.

▶ EViews Command (type in the Command window)

cpi.uroot(adf, trend, dif=1, info=sic, maxlag=13)

✎ How to interpret this output

The statistic (-7.7642) is far more negative than the 1% critical value (-4.0254) and the p-value is 0.0000. We reject the null: the first difference of CPI is stationary.

Conclusion of Step 3: CPI is integrated of order one, , so in the ARIMA(p, d, q) model. EViews will do the differencing for us inside the equation by using d(cpi).

Practical notes: (i) after differencing, the trend term is no longer needed, and a constant-only specification is normally used for D(CPI); with a statistic as extreme as -7.76 the conclusion does not change. (ii) You can repeat the test with the Phillips-Perron test (Test type: Phillips-Perron) or the KPSS test (null: stationarity) as a robustness check.

>_ EViews Command (type in the Command window)

series dcpi = d(cpi)      ' first difference of CPI
dcpi.line                 ' line graph of the differenced series
dcpi.correl(36)           ' correlogram of the differenced series

▶ Alternative: do it from the menu without creating a new series.

1. Open cpi and click View > Correlogram….
2. Select 1st difference under Correlogram of, keep 36 lags, and click OK.

✎ How to interpret this output

With 141 observations, the approximate 95% confidence band for a single autocorrelation is . Bars or values outside this band are significant.

• AC: lag 1 is clearly significant (0.391) and lag 3 is significantly negative (-0.202). Most other lags are inside the band.
• PAC: lag 1 is significant (0.391) and lag 2 is also significant (-0.186); lag 3 (-0.156) is borderline.
• Q-statistics: all p-values are small (0.000 to 0.025), so D(CPI) is clearly autocorrelated and there is something for an ARMA model to explain.

The pattern does not match a single textbook process. Recall the rules: if the PACF cuts off and the ACF tails off, think AR; if the ACF cuts off and the PACF tails off, think MA. Here the PACF points to a short AR term (lag 1), while the ACF points to an MA term (lag 1 and/or lag 3).

Candidate models: we do not try to guess a single correct answer. Instead we shortlist two plausible models and let the estimation step decide:

• ARIMA(1, 1, 1): AR(1) and MA(1) terms.
• ARIMA(1, 1, 3): AR(1) and MA(3) terms (the MA(3) term captures the lag-3 spike).

▶ EViews Steps (menu path)

1. Go to Quick > Estimate Equation….
2. In Equation specification type: d(cpi) c ar(1) ma(1)
3. Method: LS – Least Squares (NLS and ARMA). Set the Sample to 2009m03 2020m11.
4. Open the Options tab. In the ARMA box set Method to ML (maximum likelihood).
5. Click OK. When the output appears, click Name and save the equation as eq_111.

▶ EViews Command (type in the Command window)

smpl 2009m03 2020m11
equation eq_111.ls(arma=ml) d(cpi) c ar(1) ma(1)

▶ EViews Steps (menu path)

1. Go to Quick > Estimate Equation….
2. In Equation specification type: d(cpi) c ar(1) ma(3)
3. Keep the same sample and the same ML option as before, click OK, and name the equation eq_113.

▶ EViews Command (type in the Command window)

equation eq_113.ls(arma=ml) d(cpi) c ar(1) ma(3)

' For comparison, the full ARIMA(1,1,3) with all MA lags would be:
' equation eq_113full.ls(arma=ml) d(cpi) c ar(1) ma(1) ma(2) ma(3)

✎ How to interpret this output

Header block. Dependent Variable: D(CPI) confirms the differencing. Method: ARMA Maximum Likelihood shows the ML option was applied. Sample 2009M03 2020M11, 141 observations shows one observation was lost to differencing. Convergence achieved after 32 (or 29) iterations tells you the numerical optimiser found a solution; if you ever see “convergence not achieved”, do not trust the output (see the troubleshooting section).

The constant C. In an EViews ARMA equation, C is the mean of the differenced series, not an intercept in the usual regression sense. Notice that C (0.34) is almost identical to Mean dependent var (0.3412). The interpretation is: CPI rises on average by about 0.34 index points per month. It is highly significant in both models.

AR(1) and MA terms. These are the dynamics around the mean. In ARIMA(1,1,1) neither AR(1) (p = 0.2545) nor MA(1) (p = 0.1930) is significant, so the model captures very little beyond the average change. In ARIMA(1,1,3), AR(1) = 0.413 (p = 0.0000) says that a month with an above-average CPI increase tends to be followed by an above-average increase (positive persistence in monthly changes), and MA(3) = -0.299 (p = 0.0011) says that part of a shock is reversed three months later. Both are strongly significant.

SIGMASQ. This is the estimated variance of the white-noise error, . It is lower for ARIMA(1,1,3) (0.2095 versus 0.2207), meaning smaller unexplained variation.

Inverted roots. The inverted AR and MA roots must have modulus below one for the model to be stationary and invertible. All of them do here (0.20, -0.24 in the first model; 0.41, 0.67 and -0.33 ± 0.58i in the second).

Durbin-Watson values (1.97 and 1.94) are close to 2, a first hint of no first-order autocorrelation in the residuals. The formal check follows in Step 7.

Writing the fitted ARIMA(1,1,3) model. Because C is the mean of the differenced series, the estimated model is

▶ Compare the models (menu path)

1. Open each saved equation (eq_111 and eq_113) by double-clicking it in the workfile window.
2. Read the Akaike info criterion, Schwarz criterion, Hannan-Quinn criter. and Adjusted R-squared at the bottom of each output, and check the Prob. column of the coefficients.

▶ EViews Command: store the criteria as scalars (double-click a scalar to see its value)

scalar aic_111  = eq_111.@aic
scalar aic_113  = eq_113.@aic
scalar sbc_111  = eq_111.@schwarz
scalar sbc_113  = eq_113.@schwarz
scalar hqc_111  = eq_111.@hq
scalar hqc_113  = eq_113.@hq
scalar arsq_111 = eq_111.@rbar2
scalar arsq_113 = eq_113.@rbar2

✎ How to decide

Smaller values of AIC, SBC and HQC are better, and a larger adjusted R-squared is better. ARIMA(1,1,3) wins on every criterion, and all of its coefficients are significant, whereas in ARIMA(1,1,1) the ARMA terms are insignificant. We therefore select ARIMA(1,1,3) with AR(1) and MA(3) and move on to check whether it is adequate.

Overfitting check (optional): estimate the model with one extra term, for example d(cpi) c ar(1) ma(1) ma(2) ma(3). If the additional MA terms are insignificant and the criteria do not improve, the restricted model is sufficient.

▶ EViews Steps (menu path)

1. In the eq_113 window click View > Residual Diagnostics > Correlogram – Q-statistics….
2. Set Lags to include to 36 and click OK.

▶ EViews Command (type in the Command window)

eq_113.correl(36)

✎ How to interpret this output

AC and PAC columns. All residual autocorrelations and partial autocorrelations are small and lie inside the approximate band. The largest are at lags 23 and 24 (-0.137 and -0.149), which are still inside the band.

Q-stat and Prob columns. The header reads Q-statistic probabilities adjusted for 2 ARMA terms: EViews has corrected the degrees of freedom for the two ARMA terms in the model (AR(1) and MA(3)). That is why the first two lags have no probability: with two ARMA terms there are no degrees of freedom left at lags 1 and 2. From lag 3 onwards the p-values range from 0.38 to 0.95 and are all well above 0.05.

Decision: we do not reject . The residuals behave like white noise, so the model has captured the linear dependence in D(CPI). If you had found small p-values (below 0.05) at several lags, you would return to Step 4 or 5 and add or change AR or MA terms.

▶ EViews Steps (menu path)

1. In the eq_113 window click View > ARMA Structure….
2. Select Roots and, under Display, choose Graph (or Table for the numbers) and click OK.

✎ How to interpret this output

The graph plots the inverse AR roots (hollow markers) and inverse MA roots (red dots) against the unit circle. The model is stable (stationary) if the inverse AR roots lie inside the circle, and invertible if the inverse MA roots lie inside the circle.

In our model the AR root (0.41) and all three MA roots (0.67 and -0.33 ± 0.58i) are inside the circle. The model is stationary and invertible. Roots very close to the circle (modulus near 1) would be a warning sign of a near unit root or of over-differencing.

▶ Optional checks (menu path)

1. Normality: in the equation window, click View > Residual Diagnostics > Histogram – Normality Test. A Jarque-Bera p-value above 0.05 supports normal errors (important for the width of forecast bands).
2. ARCH effects: click View > Residual Diagnostics > Heteroskedasticity Tests…, choose ARCH, and click OK. A p-value below 0.05 indicates volatility clustering and suggests a GARCH-type extension.

▶ EViews Command: residual checks

eq_113.makeresids res_113     ' save the residuals as a series
res_113.hist                  ' histogram and Jarque-Bera test
res_113.line                  ' plot the residuals to look for outliers or breaks

✎ Final remark on diagnostics

Always look at the residual plot as well. A cluster of large residuals around a specific date (for example, the unusual dip in CPI around 2020) may call for a dummy variable or a more cautious reading of the forecasts.

Conclusion of Step 7: ARIMA(1,1,3) passes the residual autocorrelation test and the stability test, so it is adequate for forecasting.

▶ EViews Steps (menu path)

1. Make sure the workfile range has been extended to 2021m11 (see the setup section).
2. In the eq_113 window click Proc > Forecast… (or the Forecast button).
3. Series to forecast: choose the level (CPI) rather than D(CPI). In Forecast name type cpiforecast; optionally give a name for the standard errors such as cpiforecast_se.
4. Method: select Dynamic forecast.
5. Forecast sample: 2020m12 2021m11.
6. Under Output tick Forecast graph and Forecast evaluation, then click OK.

▶ EViews Command: plot actual and forecast together

' Graph actual CPI together with the forecast series
group g_fc cpi cpiforecast
g_fc.line

✎ How to interpret this output

The actual series (red) ends in 2020M11 at a little over 260. The forecast (blue) continues as a nearly straight upward line and reaches about 264 by 2021M11, which is about 4 index points more than the last observation. This is the expected behaviour of an ARIMA model with and a non-zero mean: the forecast of the differenced series converges quickly to its mean (about 0.34 per month), so the forecast of the level rises along a straight line with that slope. The model therefore forecasts the trend, not the short-run ups and downs.

The widening bands in the forecast graph (plus or minus two standard errors) show that uncertainty increases with the forecast horizon.

✎ How to interpret the forecast evaluation

• RMSE, MAE: average size of the forecast error in the units of CPI (index points). RMSE punishes large errors more. MAPE is the same in percent: about 0.78% for ARIMA(1,1,3). Smaller is better, and ARIMA(1,1,3) is smaller on all of them. Note that RMSE and MAE are in the units of the series, so they can only be compared between models for the same series.
• Theil inequality coefficient lies between 0 and 1; values near zero (0.0046 here) indicate a good fit between forecast and actual.
• Bias, variance and covariance proportions add up to one. The bias proportion measures how far the mean of the forecast is from the mean of the actual series, the variance proportion measures how far the variability of the forecast is from that of the actual, and the covariance proportion is the remaining, unsystematic part. A good forecast has small bias and variance proportions and a large covariance proportion. ARIMA(1,1,3) (0.17, 0.10, 0.73) is better in this respect than ARIMA(1,1,1) (0.28, 0.08, 0.65).
• Theil U2 compares the forecast with a naive “no-change” forecast; a value above 1 means the model forecasts worse than the naive one. The values here (3.58 and 3.81) are well above 1. This is not surprising for a dynamic forecast over a very long horizon, because the straight trend line cannot follow the month-to-month movements that a naive one-step forecast tracks. It is a reminder that these statistics describe trend tracking, not short-term accuracy.

Verdict: ARIMA(1,1,3) has the lower RMSE, MAE and MAPE and the more favourable error decomposition, so it is also the better forecasting model of the two.

>_ EViews Command (type in the Command window)

' 1. Estimate both models on the shorter sample
smpl 2009m03 2019m11
equation eq_111_h.ls(arma=ml) d(cpi) c ar(1) ma(1)
equation eq_113_h.ls(arma=ml) d(cpi) c ar(1) ma(3)

▶ Forecast the hold-out period (menu path)

1. Open eq_113_h, click Proc > Forecast…, forecast the level of CPI, name it cpif_113h, choose Dynamic forecast, and set the Forecast sample to 2019m12 2020m11.
2. Repeat for eq_111_h with the forecast name cpif_111h.
3. Read the evaluation box (RMSE, MAE, MAPE) for each, or compute them with the commands below.

>_ EViews Command: hold-out accuracy measures

' 2. Compute the accuracy measures over the hold-out period
smpl 2019m12 2020m11
series e_113 = cpi - cpif_113h
series e_111 = cpi - cpif_111h

scalar rmse_113 = @sqrt(@sumsq(e_113)/@obs(e_113))
scalar mae_113  = @sum(@abs(e_113))/@obs(e_113)
scalar mape_113 = 100*@sum(@abs(e_113/cpi))/@obs(e_113)

scalar rmse_111 = @sqrt(@sumsq(e_111)/@obs(e_111))
scalar mae_111  = @sum(@abs(e_111))/@obs(e_111)
scalar mape_111 = 100*@sum(@abs(e_111/cpi))/@obs(e_111)

smpl 2009m03 2020m11            ' restore the full sample

✎ How to read the hold-out results

The model with the smaller RMSE, MAE and MAPE over the hold-out months is the better genuine forecaster. If a simple benchmark, such as “next month equals this month plus the average change”, does as well as the ARIMA model, there is little reason to prefer the more complex model. Also keep in mind that unusual episodes (such as the 2020 dip) fall in the hold-out window, so differences between models in that window can reflect one-off events rather than modelling quality. When the differences are small, a formal test such as the Diebold-Mariano test is advisable before declaring a winner.

>_ EViews Program

' ==========================================================
' Box-Jenkins (ARIMA) workflow in EViews: CPI example
' ==========================================================

' ---- 0. Setup
wfopen "C:\data\cpi.xlsx"
pagestruct(end=2021m11)
smpl 2009m02 2020m11

' ---- 1. Identification
cpi.line
cpi.correl(36)
cpi.uroot(adf, trend, info=sic, maxlag=13)           ' level
cpi.uroot(adf, trend, dif=1, info=sic, maxlag=13)    ' first difference
series dcpi = d(cpi)
dcpi.line
dcpi.correl(36)

' ---- 2. Estimation
smpl 2009m03 2020m11
equation eq_111.ls(arma=ml) d(cpi) c ar(1) ma(1)
equation eq_113.ls(arma=ml) d(cpi) c ar(1) ma(3)

scalar aic_111 = eq_111.@aic
scalar aic_113 = eq_113.@aic
scalar sbc_111 = eq_111.@schwarz
scalar sbc_113 = eq_113.@schwarz
scalar hqc_111 = eq_111.@hq
scalar hqc_113 = eq_113.@hq
scalar arsq_111 = eq_111.@rbar2
scalar arsq_113 = eq_113.@rbar2

' ---- 3. Diagnostics (chosen model: eq_113)
eq_113.correl(36)
eq_113.makeresids res_113
res_113.hist

' ---- 4. Forecasting: use Proc > Forecast on eq_113 (level of CPI, dynamic,
'         sample 2020m12 2021m11), then:
group g_fc cpi cpiforecast
g_fc.line