Original Time Series:

 

Visualized the original ‘Total Gross Earnings’ over time.

Checked for any discernible trends or patterns.

 

Box-Cox Transformation:

Applied the Box-Cox transformation to stabilize variance.

Visualized the transformed time series.

 

Augmented Dickey-Fuller Test (Box-Cox Transformed):

 

Conducted the ADF test on the Box-Cox transformed series.

ADF Statistic: -4.073 (approx.)

p-value: 0.0015 (approx.)

Conclusion: The Box-Cox transformed series is stationary.

 

Differencing (Box-Cox Transformed):

 

Applied differencing to further ensure stationarity.

Visualized the differenced Box-Cox transformed series.

 

Augmented Dickey-Fuller Test (Differenced Box-Cox Transformed):

 

Conducted the ADF test on the differenced Box-Cox transformed series.

ADF Statistic: -15.524 (approx.)

p-value: < 0.001

Conclusion: The differenced Box-Cox transformed series is stationary.

Summary:

 

The Box-Cox transformation and differencing made the time series stationary, a crucial condition for many time series analyses and forecasting models. The ADF tests support the achieved stationarity.

Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF) Analysis:

Autocorrelation Function (ACF) Plot:

 

Lag Analysis: Examined autocorrelation at different time lags (up to 20 years).

Observations:

Significant positive autocorrelation at lag 1.

Gradual decline in autocorrelation as lag increases.

Interpretation:

Indicates a moderate-long term persistence or trend in the time series.

Partial Autocorrelation Function (PACF) Plot:

 

Lagged Correlation Analysis: Assessed partial correlation at various lags (up to 20 years).

Observations:

Significant partial correlation at lag 1, negligible partial correlation at subsequent lags.

Interpretation:

Suggests a direct influence of the immediate preceding year on the current year.

Summary:

 

ACF: Indicates a general declining trend in autocorrelation, signifying a weakening persistence in the time series.

PACF: Highlights a strong correlation with the immediate preceding year, potentially implying a short-term influence on the current year.

Implications for Time Series Analysis:

 

The time series may exhibit some degree of trend or persistence, requiring appropriate modeling techniques.

Immediate lag (1-year) appears influential, suggesting a potential yearly pattern or dependency.

Consideration of lagged values in forecasting models may be beneficial for capturing temporal dependencies.