Interpret the key results for ARIMA

Complete the following steps to interpret an ARIMA analysis. Key output includes the p-value, coefficients, mean square error, Ljung-Box chi-square statistics, and the autocorrelation function of the residuals.

Step 1: Determine whether each term in the model is significant

To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that the term is not significantly different from 0, which indicates that no association exists between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that the term is not significantly different from 0 when it is significantly different from 0.
P-value ≤ α: The term is statistically significant
If the p-value is less than or equal to the significance level, you can conclude that the coefficient is statistically significant.
P-value > α: The term is not statistically significant
If the p-value is greater than the significance level, you cannot conclude that the coefficient is statistically significant. You may want to refit the model without the term.
Final Estimates of Parameters Type Coef SE Coef T-Value P-Value AR 1 -0.504 0.114 -4.42 0.000 Constant 150.415 0.325 463.34 0.000 Mean 100.000 0.216
Key Results: P, Coef

The autoregressive term has a p-value that is less than the significance level of 0.05. You can conclude that the coefficient for the autoregressive term is statistically significant, and you should keep the term in the model.

Step 2: Determine how well the model fits the data

Use the mean square error (MS) to determine how well the model fits the data. Smaller values indicate a better fitting model.

Residual Sums of Squares DF SS MS 58 366.733 6.32299 Back forecasts excluded
Key Results: MS

The mean square error is 6.323 for this model. This value is not very informative by itself, but you can use it to compare the fits of different ARIMA models.

Step 3: Determine whether your model meets the assumption of the analysis

Use the Ljung-Box chi-square statistics and the autocorrelation function of the residuals to determine whether the model meets the assumptions that the residuals are independent. If the assumption is not met, the model may not fit the data and you should use caution when you interpret the results.
Ljung-Box chi-square statistics
To determine whether the residuals are independent, compare the p-value to the significance level for each chi square statistic. Usually, a significance level (denoted as α or alpha) of 0.05 works well. If the p-value is greater than the significance level, you can conclude that the residuals are independent and that the model meets the assumption.
Autocorrelation function of the residuals
If no significant correlations are present, you can conclude that the residuals are independent. However, you may see 1 or 2 significant correlations at higher order lags that are not seasonal lags. These correlations are usually caused by random error instead and are not a sign that the assumption is not met. In this case, you can conclude that the residuals are independent.

ARIMA Model: Trade

Estimates at Each Iteration Iteration SSE Parameters 0 543.908 0.100 90.090 1 467.180 -0.050 105.068 2 412.206 -0.200 120.046 3 378.980 -0.350 135.024 4 367.545 -0.494 149.372 5 367.492 -0.503 150.341 6 367.492 -0.504 150.410 7 367.492 -0.504 150.415 Relative change in each estimate less than 0.001
Final Estimates of Parameters Type Coef SE Coef T-Value P-Value AR 1 -0.504 0.114 -4.42 0.000 Constant 150.415 0.325 463.34 0.000 Mean 100.000 0.216

Number of observations: 60

Residual Sums of Squares DF SS MS 58 366.733 6.32299 Back forecasts excluded
Modified Box-Pierce (Ljung-Box) Chi-Square Statistic Lag 12 24 36 48 Chi-Square 4.05 12.13 25.62 32.09 DF 10 22 34 46 P-Value 0.945 0.955 0.849 0.940
Key Results: P-Value, ACF of Residuals

In these results, the p-values for the Ljung-Box chi-square statistics are all greater than 0.05 and none of the correlations for the autocorrelation function of the residuals are significant. You can conclude that the model meets the assumption that the residuals are independent.

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