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Introduction: Trends and Forecasting Models for Pepsi Max Dividend Returns
The analysis examines the annual dividend returns of Pepsi Max from 2010 to 2022, which is utilizing different models of forecasting.
Analysis
a) Data Description and Plotting
Figure 1: Data description and plot
(Source: Self-created in MS Excel)
The trend that is noticed in the image above, is related to the annual dividend returns for Pepsi-Max shares. In this trend, a steady increase is noticed in dividends from 2010 to 2020, with the value increasing from $0.86 to $2.26 throughout the period. Consistent year-after-year growth is represented through this in dividends for a decade. However, in 2021, there is noticed a major drop, where the decrease in dividends is noticed sharply to $1.22, which is less than half of the value of the previous year (Quan, 2020). This can indicate a major event in finance and a change in strategy for the company. In 2022, a slight recovery is noticed, with an increasing dividend of $1.34, but this is still majorly below the peak noticed in 2020. The pattern overall indicates a long growth period, which is followed by a present dramatic shift in the circumstances of finance, and policy of dividend.
b) Linear Trend Forecasting Equation
Figure 2: Linear trend forecasting equation
(Source: Self-created in MS Excel)
From the image above, it is noticed that the linear trend forecasting equation is Y=0.0731603x+0.7902764. The standard linear form is followed in this equation, which is Y=mx+b, where Y is the predicted annual dividend, x is the time variable or year, m is 0.0731603, which is the slope, or the coefficient of time, and b is 0.7902764, or y-intercept. The notation of this formula is Y=0.0731603*time+0.7902764.
It is indicated through this equation that, on average, the annual dividend is expected to increase by $0.0732 every year approximately, starting from the theoretical base value of $0.7903 when time is zero. It is indicated through the slope, a positive trend, which indicates the overall upward movement in dividends over time (Pedersen, 2020). The y-intercept is 0.7902764, which indicates the estimated dividends starting points.
However, it is needed to be noted that, this linear model does not account for the present major drop in observed dividends in 2021 to 2022. It is assumed in this model a constant rate of change which might not show accurately the behavior of future dividends in terms of the present deviation from long-term trend.
c) Quadratic Trend Forecasting Equation
Figure 3: Quadratic Trend forecasting equation
(Source: Self-created in MS Excel)
The Quadratic Trend Forecasting Equation in the figure regression analysis to model a time series with a quadratic trend. It consists of an intercept, a linear parameter with time talking about the time and a squared time parameter; time2. The Multiple R value is 0. The coefficient of determination of 0. 826 tells a good relationship between the actual and estimated values and the coefficient of multiple determination is also significant with a value of 0.684; and that illustrates that, on average, 68 females are incarcerated per each one hundred prisoners in the USA. As is expected from the inclusion of only control variables the current model explains 4% of the variance in the dependent variable (Matta, 2022). The F-statistic of 10. 807 indicating non-significant difference that supports the finding that the model has a good fit and that the relationship between the variables is statistically significant – in other words, the variables cannot be assumed to be related by chance.
The coefficients signify the part played by each term on the dependent variable. The intercept is positive as well as significant, and the sign of the linear time element is positive, indicating that the trend is positive. However, while the coefficient of the time term is positive suggesting that income increases over time, their coefficient suggests that the growth rate decreases over time as depicted below, Eugene Fama argues that this has a quadratic time trend for the following reasons. In conclusion, all the employed model gives a good account of the overall trend and variance in the data is well captured by the model.
d) Exponential Trend Forecasting Equation
Figure 4: Exponential trend forecasting equation
(Source: Self-created in MS Excel)
The Exponential Trend Forecasting Equation shown in the figure is applied to a time series which has an exponential pattern of growth. As with a majority of the models, this one also uses an intercept and a time variable to forecast future values. The Multiple R is 0 The Coefficient of determination: R a square is 0. 675 points to the fact that the actual values and the predicted values are moderately related; the R Square of 0. 454 reveals that approximately 45 per cent of. On average, this model accounts for 5% of the total variation in the dependent variable. As this is a lower ratio than in the linear or quadratic model it is still regarded as a significant relationship.
The F-statistic of 9. 149, and the p-value of 0. 011, indicate that the model is statistically significant meaning that the time variable has a significant role towards prediction. The coefficient for the time variable (0. 022) is positive and the t-statistic is 3. 41 which suggests that the dependent variable increases exponentially with time (Chansri, 2020). The intercept is close to zero and not significant, thus suggesting that the exponential growth starts from insignificance. By and large,e the model fits the exponential trend though with moderate fitness.
e) Auto-Regressive Model Selection
Figure 5: Autoregressive model selection
(Source: Self-created in MS Excel)
This is confirmed by the analysis of the model, according to which the lag1 coefficient can be dubbed statistically significant (p = 0, 00951837). The model explains about fifty per cent of the variability in these outcomes among the students. It is a moderate level of explained variation (R-squared of 0. 57) of the dependent variable. Although we could perform 2-lag and 3-lag analysis, the significant value of the first lag and lack of valuable information to compare with other models is a sign to conclude that the simple 1-lag ARIMA model is most favoured (Sahana, Yogesh, & Shalini, 2023). There are also other diagnostic checks such as autocorrelation tests, and heteroscedasticity to check the adequacy of the model.
f) Residual Analysis
Figure 6: Residual analysis and combined residual plot
(Source: Self-created in MS Excel)
From the image above it is noticed that a combined residual plot is created for models b to e. In this regard, the residuals are plotted for every model against the number that is observed. Regarding the findings, and process a discussion is done in this regard.
For every model residuals are extracted, and these residuals are linear trend b, quadratic trend c, and exponential smoothing d.
A combined plot is created with observed numbers on the x-axis, and residuals on the y-axis, using different markers, and colors for every model.
Now the combined plot is discussed. In terms of a linear trend, in the negative, and middle residuals with larger positive residuals, a pattern is demonstrated at the ends, which is indicating a non-linear relationship (Sileeporn, 2021). In terms of quadratic trend, the improvement is noticed in the linear model, with smaller residuals overall, but still demonstrates some pattern. In terms of exponential smoothing, the smallest residual presence is noticed, which is indicating it might be the best fit among these models.
g) Standard Error and R² Analysis
Figure 7: Standard error and R-square analysis
(Source: Self-created in MS Excel)
In the image above it is noticed that in terms of R-square the linear trend forecasting is 0.4076, and in terms of SYX Standard Error this value is 0.3589. In terms of R-square, the quadratic trend forecasting is present at 0.6837, and in terms of SYX Standard Error this value is 0.2750. In terms of R-square, exponential trend forecasting is present at 0.4541, and SYX standard Error is present at 0.1018 (Smith, 2023). The highest R-square is noticed regarding the quadratic model, which indicates a better fit, while the exponential model has the lowest standard error, which indicates the most accurate predictions among the models.
h) Model Selection for Forecasting
Depending on the results of the exponential trend forecasting model preferred. It has the lowest standard error at 0.1018, which indicates the most accurate predictions, despite lower value is noticed of R-square than the quadratic model. The simplicity of exponential model, and lower error makes it more considering parsimony for forecasting.
i) Forecasting for 2023
Figure 8: Forecasting for 2023
(Source: Self-created in MS Excel)
From the image above, it is noticed, that for 2023, the forecasted dividends using the quadratic trend model, which is approximately present at 1.359. This value includes the 0.334 intercepts, 4.584 linear component value, and -3.560 quadratic adjustment. The quadratic component influences the forecasts, which highlights the sensitivity of the model to time squared, which provides a nuanced prediction.
Discussion and Conclusion
Overview of Results
The data involves paid annual dividends and the logarithm of paid annual dividends for the purpose of learning growth rates. One can find spaces for such temporal variables that are used in a number of trends forecasting models. The file presents two main models: the first one – linear trend model; the second one – quadratic trend model.
It mainly involves the use of a linear trend model, with forecasts of future dividends based on a straight line growth rate assumption (Stürner, 2021). On the other hand, the quadratic trend model assumes not only the variable but also the rate of change of the variable in the course of time, either with an increase or a decrease, which in turn, can offer much better predictive accuracy if, in fact, the growth in dividends is not constant.
The efficiency of these models is given by the R coefficients or the square of multiple determination (R²), which is the indication of how accurately your future model fits past data. The value of R² that has been calculated for the quadratic model is higher that for the linear one and is 0. 6837 whereas for the linear model R² is equal to 0 .4076, and above hence making the quadratic model more appropriate to the data set.
Strengths and Limitations
The quadratic model proves to have a great significance having an R-squared of 0. It can be compared with the values given by the model 6837 that describes the overall tendency and takes into consideration the nature of non-stationary data. Its time-squared dependence gives a subtle perspective for the possibility of long-term dividend stability (Sutarmi, 2023). The exponential model has a lower value of R-squared score of 0.4541 has the lowest value of standard error (0. 1018) thus giving better predictions of future values. While being simple and accurate it is often used for forecasting but the simplicity might hide some of the trend’s intricacies.
Concluding Remarks
The matter is that the quadratic model describes the trend of dividends in detail while the exponential model is better to be used for practical forecasting as it contains minimal standard error. The quadratic model forecasts for the year 2023 the dividend would be of around $1. 359. Future studies should incorporate data up to the period of analysis and take into consideration external factors that may affect dividend payment in order to provide qualitative expectations.
