Ljung-Box Test Hypotheses:

For the Ljung-Box test, the alternative hypothesis (H1) asserts significant autocorrelation at least at one lag within the specified maximum lag, while the null hypothesis (H0) posits the absence of autocorrelation in a time series at lags up to a specified maximum lag.

Test Statistic and Critical Values:
Based on the sum of squares of autocorrelations at various lags, the test statistic is calculated. Next, the test statistic and critical values from the chi-square distribution are contrasted. There is significant autocorrelation if the test statistic is greater than the critical value, rejecting the null hypothesis.

Interpretation of Significant Autocorrelation:
A statistically significant autocorrelation present in 25% of the data indicates that the residuals cannot be adequately explained by the model. This suggests that there may be lags where the residuals are not independent or random, exposing a temporal structure that the model is unable to explain.

Implications and Model Enhancement:
Considerable autocorrelation indicates patterns or subtleties that have not yet been identified in the time series, which may be the result of factors that have been missed or underlying complexity. This emphasises how important it is to improve the model, which leads to investigating various specifications, modifying parameters, or adding new features.

Analysis of Individual Lags:
In order to identify trends and inform iterative model improvements, it becomes imperative to examine individual lags with notable autocorrelations. This in-depth analysis helps to reveal details about the data structure that the first model was unable to sufficiently represent.