MGT 267 PROJECT. Forecasting the United States Retail Sales of the Pharmacies and Drug Stores. Done by: Shunwei Wang & Mohammad Zainal

Transcription

1 MGT 267 PROJECT Forecasting the United States Retail Sales of the Pharmacies and Drug Stores Done by: Shunwei Wang & Mohammad Zainal Dec. 2002

2 The retail sale (Million) ABSTRACT The present study aims at forecasting the pharmacy and drug store retail sales in US. Different forecasting techniques are examined in the present study namely the moving average, simple exponential smoothing, Holt s exponential smoothing, Winters exponential smoothing, simple regression, multiple regression, time series decomposition and ARIMA model. Quarterly data are used to predict the retail sales using the above mentioned models. The forecast results obtained by the ARIMA are found to be the best among other models. The assessment criteria are based on the minimum RMSE, MAPE, and maximum R INTRODUCTION The retail sales of pharmacies and drug stores in the US represent essential economical data for the Pharmaceutical companies. It has a significant impact on the market decisions made by the mangers to predict future sales, inventory needs, personnel requirements, and other important economic or business forecasting. However, there are many variables that may affect forecasting of retail sales. Therefore, we are interested in forecasting the retail sales of pharmacies and drug stores in the US, and want to build up a possible forecasting model. Monthly and quarterly data of the real economic variable are obtained from the following source: ( The monthly data are arranged in quarterly format in the present investigation. Forty quarters data points from 1992 to 2001 are utilized. The retail sales of pharmacies and drug stores in the US Q1-92 Q3-92 Q1-93 Q3-93 Q1-94 Q3-94 Q1-95 Q3-95 Q1-96 Q3-96 Q1-97 Time Q3-97 Q1-98 Q3-98 Q1-99 Q3-99 Q1-00 Q3-00 Q1-01 Q3-01 It is clearly evident from the time series plot that there are certain characteristics in the retail sales of pharmacies and drug stores in the US from 1992 to These aspects can be summarized as follows 1. There is a positive trend in the above time series plot. As such there an upward movement in the pattern due to an increase in the population and health care standards. Accordingly significant amount of money is spent. Moreover, the recent advancements in the field of Pharmacy led to the development of more effective and expensive drugs compared with conventional ones. 2. A seasonal pattern occurs in the data. There is a significant increase of the retail of sail in the fourth quarter. The reasons are expected due to the followings: An increase in the cold and flu diseases is noticed in this quarter.

3 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct-01 Fourth quarter is the holidays season as such the pharmaceutical products and some other related ones are largely purchased as gifts compared to the other quarters. In general, due to globalization, companies nowadays are involved in many other types of business. One company may invest in another sister company and the whole retail of the company takes effect at the fourth quarter. The used data are separated into two groups. One is the historical data for the forecasting model, with 36 periods from Q to Q4 2000; another is holdout to test the goodness of the fit, with 4 periods from Q to Q FORECASTING TECHNIQUES AND THEIR RESULTS: 2.1 Moving Average Moving average technique is used as a forecast model for the retail sales data. Four-quarter moving average is invoked since the seasonal pattern occurs every four quarters. The US Retail Sales: Pharmacies and Drug Stores Series 1 Forecast of Series 1 Fitted Values Method 4-Quarter Moving Average Mean Absolute Percentage Error (MAPE) 4.35% R-Square 89.54% Root Mean Square Error Historic before , RMSE / Mean Holdout Q1, 2001-Q4, % ACF Upper Limit Lower Limit

4 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct PACF Upper Limit Low er Limit 2.2 Simple Exponential Smoothing (SES) Another approach is implemented herein to forecast the pharmaceutical and drug stores retail in US using SES. The outcome of the ForecastX is shown below Y Forecast of Y Fitted Values Method Exponential Smoothing Mean Absolute Percentage Error (MAPE) 4.17% R-Square 89.54% Root Mean Square Error Historic before , RMSE / Mean Holdout Q1, 2001-Q4, % Method Statistics Value Alpha Holt s Exponential Smoothing This method can be used in order to bring the forecast values closer to the values observed if the data series exhibits a trend and seasonality. This is true for our scenario.

5 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct-01 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct-01 Series 1 Forecast of Series 1 Fitted Values Method Exponential Smoothing Mean Absolute Percentage Error (MAPE) 3.47% R-Square 94.48% Root Mean Square Error Historic before , RMSE / Mean Holdout Q1, 2001-Q4, % Method Statistics Value Alpha 0.10 Gamma Winters Exponential Smoothing This method along with the previous method is an extension of the basic smoothing model. They are used for data that exhibit both trend and seasonality. Series 1 Forecast of Series 1 Fitted Values

6 Method Exponential Smoothing Mean Absolute Percentage Error (MAPE) 1.08% R-Square 99.51% Root Mean Square Error Historic before RMSE / Mean Holdout Q1, 2001-Q4, % Method Statistics Value Alpha 0.80 Beta 0.82 Gamma 0.25 Just as stated previously, there is seasonality in the retail sale data. The seasonal index of the fourth quarter is 1.07, which has a significant increment compare with other three quarters. Season Seasonal Indices Q Q Q Q ACF Upper Limit Low er Limit PACF Upper Limit Low er Limit

7 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct-01 RS 2.5 Simple Regression We hypothesize that Personal Consumption Expenditures in Medical care (X1) is influential in determining US Retail Sales: Pharmacies and Drug Stores (Y). So we look at a scatter plot of these two variables. Linear regression model RS = PCE S = R-Sq = 94.3 % R-Sq(adj) = 94.1 % PCE From this scatter plot, it is obvious that there is a positive linear relationship between these two variables. So, simple regression method can be used here. Y Forecast of Y Fitted Values The regression equation is Y = X 1 Predictor Coef SE T P Constant X Analysis of Variance Source DF SS MS F P Regression Residual Error Total

8 RESI3 RS RESI1 Essential diagnostic check based on residual analysis is carried out as shown in the figure below. One can see an existing pattern which means that the simple regression model can not fit the data properly. To overcome this drawback, a nonlinear term may be added to the regression line. Residual Analysis FITS1 Linear regression model RS = PCE PCE**2 S = R-Sq = 95.4 % R-Sq(adj) = 95.1 % PCE Residual Analysis FITS The above figures illustrates that the addition of a quadratic term improved the model and satisfied the assumption.

9 Method Linear Regression Mean Absolute Percentage Error (MAPE) 4.99% R-Square 95.4% Root Mean Square Error Historic before , RMSE / Mean Holdout Q1,2001-Q4, % The regression equation is R-Sq = 95.4 % Y = E-02X E-09X 1 2 Analysis of Variance SOURCE DF SS MS F P Regression E E Error Total E+08 SOURCE DF Seq SS F P Linear E Quadratic E Multiple-Regression Model There are many variables that may affect forecasting of retail sales pharmacies and drug stores in the US, includes the total population, gross domestic product (GDP), personal income, personal consumption expenditures in health insurance and number of outpatient visits, etc. However, a correlation may exist between some of the proposed variables, which will result in the serious error in the forecast regression model. Three explanatory variables are chosen as: 1. X1: Personal Consumption Expenditures in Medical Care ( There is a high relationship between the retail sales of the pharmacy and drug stores with the personal consumption expenditures in medical care. Generally, this explanatory variable implicitly represents the information resulted from increasing the population and personal income. A positive correlation coefficient is expected for this variable. 2. X2: Unemployment rate ( Unemployment rate is an index for the economical condition. The monthly data of the employment rate are averaged to approximate the quarterly unemployment rate. 3. X3: Inflation in Consumer Price ( The amount of retail sale is affected by the inflation in consumer price. To forecast the retail sale of pharmacies and drug stores, this explanatory variable is incorporated in our

10 model. In the same manner, the monthly data are averaged to estimate the quarterly data of inflation in consumer price. The correlation among three explanatory variables: Correlations: X1, X2, X3 X1 X2 X X Cell Contents: Pearson correlation P-Value From the result above, there is not serious multicollinearity among these three explanatory variables. Personal Consumption Expenditures in Medical care (X1), Unemployment rate (X2) and Inflation in Consumer Price (X3) are used as the explanatory variables. The regression equation is Y = X X X 3 Predictor Coef SE Coef T P Constant X X X Given that the other two variables are in the model, X 3 is not significant in this model. The regression process is carried out again to have the regression equation as Y = X X 2 Predictor Coef SE Coef T P Constant X X Analysis of Variance Source DF SS MS F P Regression Residual Error Total Dummy variables are added to the model in order to capture the seasonality in the data. As such Q2, Q3 and Q4 are coded as follows

11 Q2=1 for all second quarters and zero otherwise Q3=1 for all third quarters and zero otherwise Q4=1 for all fourth quarters and zero otherwise The regression equation is Y = X X Q Q Q 4 Predictor Coef SE Coef T P Constant X X Q Q Q The variables Q2 and Q3 are not significant in the occurrence of the others parameters and the regression process is carried out again. This gives the equation to be Y = X X Q 4 Predictor Coef SE Coef T P Constant X X Q S = 1029 R-Sq = 97.0% R-Sq(adj) = 96.8% This makes sense because only the retail of the fourth quarter has significant impact on the retail. 2.7 Time Series Decomposition The trend cycle can be estimated by smoothing the series to reduce the random variation. 2X4 Moving Average From above 2x4 MA plot, a trend in the RS data is shown.

12 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct-01 Dec-97 Feb-98 Jun-98 Aug-98 Dec-98 Feb-99 Y-Q Jun-99 Aug-99 Fitted Values Dec-99 Feb-00 Jun-00 Aug-00 Dec-00 Above is a weighted MA Smoothing technique. From the pattern, one can find that the forecast value in the right side of curve obviously smaller than the real values. It means that there is a quickly increase of the RS in the coming year. After removing the trend and isolating the seasonal component, Exponential Smoothing is used to fit the data Y Forecast of Y Fitted Values Method Exponential Smoothing Mean Absolute Percentage Error (MAPE) 0.57% R-Square 99.77% RMSE / Mean Holdout Q1,2001-Q4, % 2.8 ARIMA Model Second-order difference is implemented to remove non-stationarity from time series.

13 Apr-92 Oct-92 Apr-93 Oct-93 Apr-94 Oct-94 Apr-95 Oct-95 Apr-96 Oct-96 Apr-97 Apr-01 Oct-01 Second-Order Differences Also seasonal differencing is used to remove the seasonal factor. Second Seasonal Difference ARIMA model is used to fit the data as Y Forecast of Y Fitted Values ARIMA (2,2,0)*(1,2,1). Method ARIMA (p,d,q)*(p,d,q) Mean Absolute Percentage Error (MAPE) 0.91% R-Square 99.36%

14 RMSE / Mean Holdout Q1,2001-Q4, % This model is good for the forecast value of the last year. Method Statistics Value Method Selected Box Jenkins Model Selected ARIMA(2,2,0) * (1,2,1) Error plot 1, , DISCUSSION The moving average method is suitable for the stationery data. However, our situation involves non-stationery data. The R-square in this model is only about 89.54% and the holdout RMSE/Mean is about 1.93%. From ACF and PACF, it is noticed that some autocorrelations are significantly different from zero (at lag 4, 8 and 12) which assures the seasonality at fourth quarter. No significant difference is found between the SES technique and the moving average technique. However, SES attained smaller holdout RMSE/Mean (SES: 1.93%, MA(4): 1.52%). Again SES is designated for a stationery data which is not true for our case. Since Holt s Exponential Smoothing adds a growth factor (or trend factor) to the equation as a way of adjusting for the trend, the model is better than former. The holdout RMSE in 2001 is reduced to 0.89% in this model. However, the seasonality factor in this model is still not considered. So, still there is a space to improve our model. For the Winter s Exponential Smoothing method, one can see that the holdout RMSE/Mean is 0.5% for the last year. Also, MAPE has significantly reduced, and R-Square is nearly 100% (99.51%). The forecast error has only 0.50% for the last year. Also, no significant autocorrelation is found for this forecasting technique. It is found in the results that the MAPE of the simple linear regression is bigger than previous forecast models (4.99%), and the R-Square for both of the simple linear and multiple linear regression are not very high yet.

15 The time series decomposition fits the historic data seems well, R-Square is 99.77%, However, the RMSE for the last year is a bit larger (1.91%). Finally, ARIMA model is evaluated using 2 nd order difference to achieve stationarity in the data. Also, 2 nd order difference is implemented to de-seasonalize the data. ARIMA model is found to have the minimum RMSE/MEAN ratio (0.24%) compared to other models. Error pattern seems to follow a white noise model. 4. CONCLUSION Different forecasting methods are utilized to predict the retail sale in US. The ARIMA technique exhibits best performance among other models. The RMSE and MAPE are found to be optimum for ARIMA (2,2,0)*(1,2,1). The table below shows the predicted values for the next two years using ARIMA model along with the holdout period for Forecast -- Box Jenkins Selected Actual Forecast Date Quarterly Quarterly Annual Mar , Jun , Sep , Dec , , Mar , Jun , Sep , Dec , , Mar , Jun , Sep , Dec , , Avg 34, , Max 37, , Min 31, , Holdout period