Load the Deutschmark/British pound foreign-exchange rate data set.

load Data_MarkPound

Convert the prices to returns.

returns = price2ret(Data);

Compute the deviations of the return series.

res = returns - mean(returns);

Test the hypothesis that the residual series is not autocorrelated, using the default number of lags.

h1 = lbqtest(res)

h1 = logical
   0

h1 = 0 indicates that there is not enough evidence to reject the null hypothesis that the residuals of the returns are not autocorrelated.

Test the hypothesis that there are significant ARCH effects, using the default number of lags [3].

h2 = lbqtest(res.^2)

h2 = logical
   1

h2 = 1 indicates that there are significant ARCH effects in the residuals of the returns.

Test for residual heteroscedasticity using archtest and the default number of lags.

h3 = archtest(res)

h3 = logical
   1

h3 = 1 indicates that the null hypothesis of no residual heteroscedasticity should be rejected in favor of an ARCH(1) model. This result is consistent with h2.

Conduct multiple Ljung-Box Q-tests for autocorrelation by including various lags in the test statistic. The data set is a time series of 57 consecutive days of overshorts from an underground gasoline tank in Colorado [2]. That is, the current overshort ($$y_t$$) represents the accuracy in measuring the amount of fuel:

Load the data set.

load('Data_Overshort')
y = Data;
T = length(y); % Sample size

figure
plot(y)
title('Overshorts for 57 Consecutive Days')

lbqtest is appropriate for a series with a constant mean. Since the series appears to fluctuate around a constant mean, you do not need to transform the data.

Compute the residuals.

res = y - mean(y);

Assess whether the residuals are autocorrelated. Include 5, 10, and 15 lags in the test statistic computation.

[h,pValue] = lbqtest(res,'lags',[5,10,15])

h = 1x3 logical array

   1   1   1

pValue = 1×3

    0.0016    0.0007    0.0013

h and pValue are vectors containing three elements corresponding to tests at each of the three lags. The first element of each output corresponds to the test at lag 5, the second element corresponds to the test at lag 10, and the third element corresponds to the test at lag 15.

h = 1 indicates the rejection of the null hypothesis that the residuals are not autocorrelated. pValue indicates the strength at which the test rejects the null hypothesis. Since all three $$p-values$$ are less than 0.01, there is strong evidence to reject the null hypothesis that the residuals are not autocorrelated.

Infer residuals from an estimated ARIMA model, and assess whether the residuals exhibit autocorrelation using lbqtest.

Load the Australian Consumer Price Index (CPI) data set. The time series (cpi) is the log quarterly CPI from 1972 to 1991. Remove the trend in the series by taking the first difference.

load Data_JAustralian
cpi = DataTable.PAU;
T = length(cpi);
dCPI = diff(cpi);

figure
plot(dates(2:T),dCPI)
title('Differenced Australian CPI')
xlabel('Year')
ylabel('CPI growth rate')
datetick
axis tight

The differenced series appears stationary.

Fit an AR(1) model to the series, and then infer residuals from the estimated model.

Mdl = arima(1,0,0);
EstMdl = estimate(Mdl,dCPI);

 
    ARIMA(1,0,0) Model (Gaussian Distribution):
 
                  Value      StandardError    TStatistic      PValue  
                _________    _____________    __________    __________

    Constant     0.015564      0.0028766        5.4106      6.2808e-08
    AR{1}         0.29646        0.11048        2.6834       0.0072876
    Variance    0.0001038     1.1932e-05        8.6994      3.3362e-18

res = infer(EstMdl,dCPI);
stdRes = res/sqrt(EstMdl.Variance); % Standardized residuals

Assess whether the residuals are autocorrelated by conducting a Ljung-Box Q-test. The standardized residuals originate from the estimated model (EstMdl) containing parameters. When using such residuals, it is best practice to do the following:

lags = 10;         
dof  = lags - 1; % One autoregressive parameter

[h,pValue] = lbqtest(stdRes,'Lags',lags,'DOF',dof)

h = logical
   1

pValue = 0.0119

pValue = 0.0130 suggests that there is significant autocorrelation in the residuals at the 5% level.