ECON30401 · Computer lab
ARMA in EViews
the UK GDP lab
The theory becomes real at the keyboard. This computer practical walks through modelling quarterly UK real GDP in EViews from start to finish: importing and plotting the data, constructing a growth rate, reading the correlogram, estimating AR, MA and ARMA models, and using the diagnostics and information criteria to pick the model the data support.
The dataset is quarterly UK real GDP, seasonally adjusted by the Office for National Statistics, spanning the mid-1950s to 2013 (workfile gdp.wf1). EViews commands are shown in a fixed font; type them into the command line and press Enter, or use the equivalent menu route. Save your output as you go by pasting into a document.
Part 1 · Getting to know the data
(a) Plot the level of GDP
Graph the series GDP (double-click the series, then View → Graph → OK). UK real GDP trends strongly upward over the whole sample — a classic non-stationary series in levels, whose mean plainly changes over time. That trend is exactly why we do not model the level directly.
(b) Construct the growth rate
Create the percentage quarterly growth rate by differencing the log and scaling by 100:
series gdpgr = 100*dlog(gdp)
Here dlog takes the change in the log, so gdpgr is approximately the percentage growth of GDP from one quarter to the next (EViews is case-insensitive, and you can drop series if you use the Genr button). Graph gdpgr: it now fluctuates around a small positive average rather than trending, i.e. differencing has removed the trend and delivered something that looks stationary.
(c) Descriptive statistics
Obtain a histogram and summary statistics for gdpgr (View → Descriptive Statistics & Tests → Histogram and Stats). Note the mean, median and standard deviation. During the global financial crisis GDP fell about 2.2% in 2008Q4 and 2.5% in 2009Q1; against the histogram and the sample standard deviation these are strikingly large negative outliers, several standard deviations below the average quarterly growth.
(d) The correlogram
Obtain the correlogram of gdpgr (View → Correlogram, leaving the selection at Level — do not difference again). The first autocorrelations are clearly positive and outside the \(\pm1.96/\sqrt{T}\) bands, then decay: growth is strongly autocorrelated, so it is not white noise. The positive, quickly-decaying pattern is the signature of a low-order autoregression.
Part 2 · Estimating time series models
(a) An AR(1)
Estimate an AR(1) for growth:
ls gdpgr c gdpgr(-1)
ls is least squares, the first variable is the dependent one, c is the constant and gdpgr(-1) is the first lag (you can also use Quick → Estimate Equation). Comment formally: are the constant and the AR coefficient significant, and does the coefficient lie inside the stationary region \(|\phi_1|<1\)? Check the residual correlogram or a Breusch–Godfrey test — a well-specified AR(1) should leave white-noise residuals.
(b) An MA(1)
Now estimate an MA(1):
ls gdpgr c ma(1)
Again comment on significance and, crucially, on the residual diagnostics. Compare with the AR(1): which model leaves cleaner (less autocorrelated) residuals?
(c) Compare ARMA candidates
Estimate one or two richer specifications — for instance an ARMA(2,1):
ls gdpgr c gdpgr(-1 to -2) ma(1)
Then choose between the candidates using two criteria together: (i) the diagnostics — the preferred model should have no significant residual serial correlation; and (ii) the information criteria, Akaike (AIC) and Schwarz (SIC), reported in every estimation output, preferring the model that minimises them. For UK GDP growth this workflow typically points to a parsimonious AR(1): its coefficients are significant, its residuals are clean, and both AIC and SIC select it over the alternatives.
Plot → difference to stationarity → read the correlogram → estimate rival AR/MA/ARMA models → test the residuals → let AIC and SIC break ties. This is the same identification-estimation-diagnostics loop set out in the estimation and model selection note.
Check your understanding
Why model 100*dlog(gdp) rather than gdp itself?
The level of GDP trends upward and is non-stationary, so the stationary time series theory (constant mean and variance, well-behaved OLS) does not apply to it. The log-difference \(100\,\Delta\log(\mathrm{GDP})\) is the percentage quarterly growth rate, which fluctuates around a stable positive mean and is (approximately) stationary — the right object to model with AR/ARMA methods.
Your estimated AR(1) has clean residuals but the ARMA(2,1) has a marginally lower AIC. Which do you report?
Prefer the parsimonious AR(1) unless the richer model is clearly better. If the AR(1) already leaves white-noise residuals and SIC (which penalises extra parameters more heavily) selects it, a marginal AIC gain for the ARMA(2,1) is weak grounds for the extra complexity — especially if its added coefficients are individually insignificant. Report the AR(1) and note the comparison.
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