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# AQ019-3-2 Forensic Accounting And Fraud Examination

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• Course Code: AQ019-3-2
• University: Asia Pacific International College
• Country: Australia

## Question:

You are required to collect a secondary time series data set on any subject that you are interested. Make sure the sample size is at least 100.

Write a report to present the results and findings of the analysis. Conduct the analysis using any of the statistical software e.g. Excel, ForecastX, Eviews, SPSS, etc. Your report should cover the following requirements:
(a)Plot the original time series and explain the time-series component that present in the data.
(b)Choose any THREE of the following techniques to forecast the sales.
(i)Moving Average
(ii)Weighted Moving Average
(iii)Double Moving Average
(v)Simple Exponential Smoothing Method
(vi)    Holts’ Method
(vii)    Winters’ Method
Provide detailed descriptions and reasons of the all three forecast methods you have chosen.
(c)    Perform and provide the forecast using the methods you selected in part (b) only for those periods which are possible (existing period). Then, provide graph to compare between observed and forecast values for each method.
(d)    Name two measures of forecast error that should be used in the above situations and provide reasons of the forecast errors chosen. Then, evaluate the statistics of forecast error for each method used in part (b). Identify which forecast method is more appropriate using the measure of forecast error and explain why.
(e)    Use the most appropriate forecast method in part (d) to forecast the next 12 period (Ex post forecast).
(f)    Conduct the autocorrelation analysis (ACF and PACF) to explore the data pattern for the data. Provide detailed descriptions of the findings from the autocorrelation analysis.
(g)    Develop a regression model that uses additive dummy variables for the seasonal component if necessary, and forecast the next 12 period. Find the measure of forecast errors using the same forecast error as in part (d). Explain the findings from the regression model.

### Introduction

For the purpose of this project time series data has been collected from world bank. The data on average monthly rainfall in Malaysia for the 9-year period from 2007 to 2015 has been collected for this study. A total of 108 data points are available for the analysis. The analysis has been conducted using the statistical software SPSS version 20.

The original time series is plotted in figure 1. From the figure, it can be seen clearly that the average monthly rainfall follows a seasonal fluctuation every year.

By seasonal fluctuations it is meant that there is a periodic movement in a time series where the period is not longer than 1 year. A periodic movement in time series is one which recurs or repeats at regular intervals of time (or periods) (Montgomery, Jennings and Kulahci 2015).

C.C. Holt first suggested this this method of forecasting in 1958. This method is usually used to forecast values which do not have any systematic trend and follows non-seasonal time series data. In reality, the data that are obtained do not usually follow any seasonal pattern. The non-seasonal effects are measurable and can be removed and thus, the developed and revised model will be stationary.

In exponential smoothing technique of forecasting, data which is older is given lesser priority and the data which are new are given more priority. This method is also known as averaging and is used to forecast values for a shorter term (Ramtirthkar et al. 2016).

The forecast using the simple exponential smoothing method for the year 2016 is given in table 2.1.

 Table 2.1: Forecast Exponential Smoothing Model Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Rainfall (in mm)-Model_1 Forecast 255.58 255.58 255.58 255.58 255.58 255.58 255.58 255.58 255.58 255.58 255.58 255.58 UCL 429.46 429.55 429.64 429.73 429.82 429.91 430.00 430.09 430.18 430.27 430.36 430.45

### Holt’s Method

This method is an extension of the simple exponential smoothing method and has been developed by Holt in 1957. Hence the name. This method allows the forecasting data along with a trend. Thus, there are more than one equations involved in this type of forecasting. One of them is the forecasting equation and two others are smoothing equations (Box et al. 2015).

The forecast using the Holt’s method for the year 2016 is given in table 2.2.

 Table 2.2: Forecast Holt’s Method Model Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Rainfall (in mm)-Model_1 Forecast 237.27 236.69 236.11 235.52 234.94 234.36 233.78 233.20 232.61 232.03 231.45 230.87 UCL 412.77 412.75 412.73 412.70 412.68 412.65 412.62 412.59 412.56 412.53 412.50 412.46 LCL 61.77 60.63 59.49 58.35 57.21 56.07 54.93 53.80 52.67 51.54 50.41 49.28

Winter’s Method

The holt’s method of forecasting was extended by Winters in 1960 to capture the seasonality in the forecast. Thus, in this case, along with the three equations that explains the forecasting and the smoothing, another extra smoothing equation has been introduced. This method has two types of variations. The additive method is used when there are roughly constant seasonal variations. The multiplicative model is useful when the changes in the seasonal variations are proportional to the level of the series. The seasonal component is supposed to add up to be zero within a year (Brockwell and Davis 2016).

The forecast using the Winter’s method for the year 2016 is given in table 2.3.

 Table 2.3: Forecast Winter’s Method Model Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Rainfall (in mm)-Model_1 Forecast 305.71 182.68 200.87 196.87 207.98 150.39 157.43 203.59 175.10 248.05 304.66 343.35 UCL 428.24 305.75 324.47 321.00 332.63 275.57 283.13 329.81 301.84 375.30 432.44 471.64 LCL 183.18 59.62 77.28 72.75 83.33 25.22 31.73 77.36 48.35 120.79 176.89 215.07

It can be easily observed from the three different types of forecasting methods that the Winter’s method of forecasting shows the most accurate forecast as the value of r square is the highest in this method as can be seen from table 2.6. This indicates that the developed Winter’s model will explain the variability in the trend by 74.3 percent. This also indicates that the error is minimum in this model.

 Table 2.4: Model Statistics Exponential Smoothing Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared Statistics DF Sig. Rainfall (in mm)-Model_1 0 .158 108.073 17 .000 0
 Table 2.5: Model Statistics Holt’s Method Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared Statistics DF Sig. Rainfall (in mm)-Model_1 0 .669 112.890 16 .000 0

 Table 2.6: Model Statistics Winter’s Method Model Number of Predictors Model Fit statistics Ljung-Box Q(18) Number of Outliers Stationary R-squared Statistics DF Sig. Rainfall (in mm)-Model_1 0 .743 23.602 15 .072 0

### Autocorrelations

It can be seen from the developed ACF that there are spikes at lags 1, 5, 6, 11, 12 and 13. All the lags are significant. Thus, it can be said that there is no existence of autocorrelation in the model developed. The ACF measures are given in table 3.1. The values also sum up to be zero.

 Table 3.1: Autocorrelations Series: Predicted value from Rainfall-Model_1 Lag Autocorrelation Std. Errora Box-Ljung Statistic Value df Sig.b 1 .650 .090 51.955 1 .000 2 .265 .090 60.691 2 .000 3 -.024 .089 60.762 3 .000 4 -.168 .089 64.330 4 .000 5 -.282 .089 74.418 5 .000 6 -.432 .088 98.357 6 .000 7 -.273 .088 108.047 7 .000 8 -.207 .087 113.635 8 .000 9 -.070 .087 114.277 9 .000 10 .198 .087 119.483 10 .000 11 .563 .086 161.999 11 .000 12 .839 .086 257.390 12 .000 13 .526 .085 295.311 13 .000 14 .191 .085 300.328 14 .000 15 -.065 .085 300.916 15 .000 16 -.193 .084 306.156 16 .000 a. The underlying process assumed is independence (white noise). b. Based on the asymptotic chi-square approximation.

The partial autocorrelation shows the correlation between any two of the lags (Granger and Newbold 2014). It can be seen from the results that the partial autocorrelation at each of the lags are quite high, it is positive at some lags and negative at some other lags. The partial correlations sum up to be zero.

 Table 3.2: Partial Autocorrelations Series: Predicted value from Rainfall-Model_1 Lag Partial Autocorrelation Std. Error 1 .650 .091 2 -.272 .091 3 -.124 .091 4 -.045 .091 5 -.192 .091 6 -.300 .091 7 .326 .091 8 -.385 .091 9 .201 .091 10 .470 .091 11 .391 .091 12 .399 .091 13 -.470 .091 14 -.122 .091 15 .106 .091 16 -.059 .091

### Regression Analysis

A regression model has been developed for the forecasting purpose. Regression has been performed on the Winter’s model as that model is the best fitted model. From the results of the regression, it can be seen that the R Square value is 0.053, which indicates that the seasonal dummies can represent only 5.3 percent of the variations in the rainfall measures. It can also be said from the coefficients that with the increase in the years and months, the average monthly rainfall decreases by 0.474.

 Table 4.1: Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 .230a .053 .044 62.93165 a. Predictors: (Constant), Seasonal_Dummy
 Table 4.2: ANOVAa Model Sum of Squares df Mean Square F Sig. 1 Regression 23551.351 1 23551.351 5.947 .016b Residual 419801.595 106 3960.392 Total 443352.946 107 a. Dependent Variable: Predicted value from Rainfall-Model_1 b. Predictors: (Constant), Seasonal_Dummy
 Table 4.3: Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 297.605 12.196 24.402 .000 Seasonal_Dummy -.474 .194 -.230 -2.439 .016 a. Dependent Variable: Predicted value from Rainfall-Model_1

## Conclusions

Analysis has been done in this project on the monthly rainfall of Malaysia. The methods of prediction that has been used here are Exponential Smoothing, Holt’s Method and Winter’s Method. It has been observed from the analysis that the Winter’s Method has been the most appropriate method of prediction as the error in variation has been the minimum in this model fit. On prediction with the help of this model, it has been seen that the error in prediction is high and the average monthly rainfall decreases with time.

## References

Box, G.E., Jenkins, G.M., Reinsel, G.C. and Ljung, G.M., 2015. Time series analysis: forecasting and control. John Wiley & Sons.

Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. springer.

Granger, C.W.J. and Newbold, P., 2014. Forecasting economic time series. Academic Press.

Montgomery, D.C., Jennings, C.L. and Kulahci, M., 2015. Introduction to time series analysis and forecasting. John Wiley & Sons.

Ramtirthkar, M., Gupta, A., Sonawane, R. and Kolhatkar, A., 2016. Forecasting yield of coarse cereals in India using ARIMA model. AgricINTERNATIONAL, 3(1), pp.117-127.

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