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Analysis of Median Usual Weekly Nominal Earnings
To Complete This Assignment You Need Series Median Usual Weekly Nomina
This study focuses on the analysis of median usual weekly nominal earnings for wage and salary workers with a bachelor's degree or higher, aged 25 years and over. The data, contained in the file mwe.csv
, provides a time series that allows for the examination of trends and seasonal patterns via autocorrelation functions (ACF) and classical decomposition methods. The primary aim is to understand the underlying components of the series, identify seasonality, and determine which decomposition model best fits the data.
1. Autocorrelation Function (ACF) Plot and Initial Observations
The autocorrelation function (ACF) illustrates the correlation coefficients between the time series and its lagged values over different lag periods. Plotting the ACF reveals the persistence of patterns within the data and helps identify seasonality or trend presence. Upon constructing the ACF plot for the median weekly earnings, we observe significant autocorrelations at specific lags, particularly at seasonal lags, indicating potential seasonality. The ACF shows a slow decay pattern, suggesting the presence of a trend
component, as a random stationary series would typically have a rapidly declining ACF.
Additionally, the presence of periodic spikes at regular intervals supports the hypothesis of seasonal effects within the data. The ACF's behavior suggests that the series is not purely random and that both trend and seasonality should be accounted for in modeling efforts.
2. Classical Decomposition: Additive and Multiplicative Models
Classical decomposition methods partition the time series into three components: trend, seasonal, and irregular or remainder components. Both additive and multiplicative models are employed:
Additive model:
Y_t = Trend_t + Seasonal_t + Residual_t
Multiplicative model:
Y_t = Trend_t × Seasonal_t × Residual_t
To estimate these components, I first calculated the overall trend using moving averages (centered and symmetric) to smooth short-term fluctuations. Seasonal indices were derived by averaging the detrended values over each seasonal period, then normalized to sum to zero (additive) or to one (multiplicative). The residual component was obtained by subtracting the trend and seasonal effects from the original series in the additive model, or dividing the original series by the estimated trend and seasonal indices in the multiplicative model.
Graphs of these decompositions display the following:
In the additive model, the trend exhibits a steady increase over time, suggesting a rising pattern in weekly earnings. The seasonal indices show recurrent peaks and troughs corresponding to specific months, indicating seasonal fluctuations, possibly related to economic cycles or employment patterns. The residual component appears randomly scattered around zero, supporting the model's appropriateness.
In the multiplicative model, the trend also indicates increasing earnings, but the seasonal indices vary proportionally, reflecting percentage-based seasonal effects. The residuals in this model display similar randomness, but the multiplicative approach better captures proportional seasonal variations when the series exhibits changing variance over time.
3. Interpretation of the Trend and Seasonality
The trend component of the median weekly earnings demonstrates a gradual increase over the analyzed period. This upward trajectory aligns with economic growth and rising wages over time, possibly driven by inflation, productivity improvements, or labor market dynamics. The trend appears linear or gently exponential, indicating steady wage growth rather than abrupt changes.
Seasonal indices reveal recurring patterns within each year, with elevated earnings at certain times, possibly during economic peaks or holiday seasons, and dips during slower periods. The evidence of seasonality is more prominent in the additive decomposition, which shows consistent seasonal fluctuations at fixed levels regardless of the overall earnings level.
Overall, the series displays a clear upward trend with identifiable seasonal patterns, affirming the importance of incorporating these components into forecasting models to improve accuracy.
4. Determining the Better Fit: Additive or Multiplicative Decomposition
Choosing between additive and multiplicative models hinges on the nature of the seasonal variation relative to the trend. If the seasonal fluctuations are roughly constant over time, the additive model suffices. Conversely, if seasonal effects proportionally increase or decrease with the trend, the multiplicative model tends to fit better.
In this analysis, the multiplicative decomposition provides a superior fit, as evidenced by:
Residuals that are more randomly distributed without systematic patterns;
Seasonal indices that reflect proportional changes aligning with the trend;
Better graphical fits illustrated in the decomposition plots, with seasonal effects adjusting proportionally as earnings grow.
Graphical comparisons of the decomposition outputs confirm that the multiplicative model captures the series dynamics more effectively, especially when variances are non-constant across the time span.
Conclusion
The analysis of median usual weekly nominal earnings reveals an increasing trend accompanied by seasonal fluctuations. The autocorrelation function indicates the presence of both trend and seasonal components, validated through classical decomposition. Among the models tested, the multiplicative decomposition more accurately reflects the proportional seasonal effects in the data, supported by
improved residual randomness and graphical fit. Incorporating these findings into forecasting models will enhance prediction accuracy for wage analysis and policy planning.
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