Identifying The Need For Advanced Regression Analysis

Other Forms of Regression . In 2-variable regression analysis, you use a single independent variable (X) to estimate the dependent variable (Y), and the relationship is assumed to form a straight line. This is the most common form of regression analysis used in contract pricing. However, when you need more than one independent variable to estimate cost or price, you should consider multiple regression (or multivariate linear regression). When you expect that a trend line will be a curve instead of a straight line, you should consider curvilinear regression. A detailed presentation on how to use multiple regression or curvilinear regression is beyond the scope of this text. However, you should have a general understanding of when and how these techniques can be applied to contract pricing. When you identify a situation that seems to call for the use of one of these techniques, consult an expert for the actual analysis. You can obtain more details on the actual use of these techniques from advanced forecasting texts. Multiple Regression Situation . Multiple regression analysis assumes that the change in Y can be better explained by using more than one independent variable. For example, suppose you want to determine the relationship between main-frame computer hours, field-audit hours expended in audit analysis, and the cost reduction recommendations sustained during contract negotiations. Computer Hours Field Audit Hours Sustained Reduction 1.4 45 $290,000 1.1 37 $240,000 1.4 44 $270,000 1.1 45 $250,000 1.3 40 $260,000 1.5 46 $280,000 1.5 47 $300,000 It is beyond the purpose of this text to demonstrate how a multivariate equation would be developed using this data. However, we will describe the elements of the multivariate equation and the results of a regression analysis. Three-Variable Linear Equation . Multiple regression can involve any number of independent variables. To solve the audit example above, we would use a three-variable linear equation -- two independent variables and one dependent variable. Y C = A + B1 X 1 + B2 X 2 Where: Yc = The calculated or estimated value for the dependent (response) variable A = The Y intercept, the value of Y when X 1 = 0 and X 2 = 0 X 2 = The first independent (explanatory) variable B2 = The slope of the line related to the change in X 1 , the value by which Y changes when X 1 changes by one. X 2 = The second independent (explanatory) variable B2 = The slope of the line related to the change in X 2 , the value by which Y changes when X 2 changes by one. Results of Audit Data Three-Variable Linear Regression AnalysisI. Using the above data on audit analysis and negotiated reductions, an analyst identified the following three variables: X 2 = Computer Hours X 2 = Field Audit Hours Y = Cost Reductions Sustained The results of analysts analysis are shown in the following table: Regression Results Predictor Variable Equation r 2 Computer Hours Y = A + BX 1 .82 Field Audit Hours Y = A + B X 2 .60 Comp Hrs and Field Audit Hrs Y = A + B1 X 1 + B2 X 2 .88 You can see from the r 2 values in the above table that computer hours explains more of the variation in cost reduction recommendations sustained than is explained by field audit hours. If you had to select one independent variable, you would likely select Computer Hours. However, the combination of the two independent variables in multiple regression explains more of the variation in cost reduction recommendations sustained than the use of computer hours alone. The combination produces a stronger estimating tool. Curvilinear Regression Analysis . In some cases, the relationship between the independent variable(s) may not be linear. Instead, a graph of the relationship on ordinary graph paper would depict a curve. You cannot directly fit a line to a curve using regression analysis. However, if you can identify a quantitative function that transforms a graph of the data to a linear relationship, you can then use regression analysis to calculate a line of best fit for the transformed data. Common Transformation Functions Examples Reciprocal Square Root Log-Log logX Power X 2 For example, improvement curve analysis (presented later in this text) uses a special form of curvilinear regression. While it can be used in price analysis and material cost analysis, the primary use of the improvement curve is to estimate labor hours. The curve assumes that less cost is required to produce each unit as the total units produced increases. In other words, the firm becomes more efficient as the total units produced increases. There are many improvement curve formulations but one of the most frequently used is: Y = AX B Where: Y = Unit cost (in hours or dollars of the Xth unit) X = Unit number A = Theoretical cost of the first unit B = Constant value related to the rate of efficiency improvement Obviously, this equation does not describe a straight line. However, using the logarithmic values of X and Y (log-log transformation), we can transform this curvilinear relationship into a linear relationship for regression analysis. The result will be an equation in the form: logY = logA + BlogX Where: logY = The logarithmic value of Y logA = The logarithmic value of A logX = The logarithmic value of X We can then use the linear equation to estimate the logarithmic value of Y, and from that Y. 5.7 - Identifying Issues And Concerns Questions to Consider in AnalysisI. As you perform price/cost analysis, consider the issues and concerns identified in this section, whenever you use regression analysis. • Does the r 2 value indicate a strong relationship between the independent variable and the dependent variable? The value of r 2 indicates the percentage of variation in the dependent variable that is explained by the independent variable. Obviously, you would prefer an r 2 of .96 over an r 2 of .10, but there is no magic cutoff for r 2 that indicates that an equation is or is not acceptable for estimating purposes. However, as the r 2 becomes smaller, you should consider your reliance on any prediction accordingly. • Does the T-test for significance indicate that the relationship is statistically significant? Remember that with a small data set, you can get a relatively high r 2 when there is no statistical significance in the relationship. The T-test provides a baseline to determine the significance of the relationship. • Have you considered the prediction interval as well as the point estimate? Many estimators believe that the point estimate produced by the regression equation is the only estimate with which they need to be concerned. The point estimate is only the most likely estimate. It is part of a range of reasonable estimates represented by the prediction interval. The prediction interval is particularly useful in examining risk related to the estimate. A wide interval represents more risk than a narrow interval. This can be quite valuable in making decisions such as contract type selection. The prediction interval can also be useful in establishing positions for negotiation. The point estimate could be your objective, the lower limit of the interval your minimum position, and the upper limit your maximum position. • Are you within the relevant range of data? The size of the prediction interval increases as the distance from increases. You should put the greatest reliance on forecasts made within the relevant range of existing data. For example, 12 is within the relevant range when you know the value of Y for several values of X around 12 (e.g., 10, 11, 14, and 19). • Are time series forecasts reasonable given other available information? Time series forecasts are all outside the relevant range of known data. The further you estimate into the future, the greater the risk. It is easy to extend a line several years into the future, but remember that conditions change. For example, the low inflation rates of the 1960s did not predict the hyper-inflation of the 1970s. Similarly, inflation rates of the 1970s did not predict inflation rates of the 1980s and 90s. • Is there a run of points in the data? A run consisting of a long series of points which are all above or all below the regression line may occur when historical data are arranged chronologically or in order of increasing values of the independent variable. The existence of such runs may be a symptom of one or more of the following problems: • o Some factor not considered in the regression analysis is influencing the regression equation (consider multivariate regression); o The equation being used in the analysis does not truly represent the underlying relationship between the variables; o The data do not satisfy the assumption of independence; or o The true relationship may be curvilinear instead of linear (consider curvilinear regression). • Have you graphed the data to identify possible outliers or trends that cannot be detected through the mathematics of fitting a straight line? When you use 2-variable linear regression, you will fit a straight line through the data. However, the value of the relationship identified may be affected by one or more outliers that should not really be considered in your analysis. These can be easily identified through the use of a graph. Remember though, you cannot discard a data point simply because it does not fit on the line. The graph will help you identify an outlier, but you cannot discard it unless there is a valid reason (e.g., different methods were used for that item). A graph can also permit you to identify situations where a single simple regression equation is not the best predictor. The graph may reveal that there is more than one trend affecting the data (e.g., the first several data points could indicate an upward trend, the latter data points a downward trend). It could also reveal the true relationship is a curve and not a straight line. • Have you analyzed the differences between the actual and predicted values? Like the graph, this analysis will provide you information useful in identifying outliers (e.g., there may be one very large variance affecting the relationship). However, the outlier may not be as easy to identify as with a graph because the line will be pulled toward the outlier. • Are you comparing apples with apples? Regression analysis, like any technique based on historical data, assumes that the past is a good predictor of the future. For example, you might establish a strong relationship between production labor hours and quality assurance labor hours. However, if either production methods or quality assurance methods change substantially (e.g., automation) the relationship may no longer be of any value. • How current are the data used to develop the estimating equation? The more recent the data, the more valuable the analysis. Many things may have changed since the out-of-date data were collected. • Would another independent variable provide a better estimating tool? Another equation may produce a better estimating tool. As stated above, you would likely prefer an equation with an r 2 of .96 over one with an r 2 of .10. • Does the cost merit a more detailed cost analysis? If the cost is high and the r 2 is low, it may merit a more detailed analysis. For example, if you had a relatively low r 2 for a production labor effort, it may be worth considering the use of work measurement techniques in your analysis. • 6.0 - Chapter Introduction • 6.1 - Identifying Situations For Use • 6.2 - Determining Which Moving Average Model To Use • 6.3 - Evaluating And Using Single Moving Averages • 6.4 - Evaluating And Using Double Moving Averages • 6.5 - Identifying Issues And Concerns 6.0 - Chapter Introduction In this chapter, you will learn to use moving averages to estimate and analyze estimates of contract cost and price. Single Moving Average . If you cannot identify or you cannot measure an independent variable that you can use to estimate a particular dependent variable, your best estimate is often an average (mean) of past observations. The single moving average builds on this principle by defining the number of observations that you will consider. It assumes that the recent past is the best predictor of the future. In a single moving average, data collected over two or more time periods (normally at least three) are summed and divided by the number of time periods. That average then becomes a forecast for future time periods. As data from a new time period is added, data from an earlier time period is dropped from the average calculation. For example, a 12-month moving average uses data from the most recent 12 months. A 6-month moving average uses data from the most recent six months. You must determine the appropriate number of time periods to consider in the analysis. You can use any time period, but monthly data is the most common. Double Moving Average. If you believe that there is a trend in the data, you can use a double moving average. A trend in the data means that the observation values tend to either increase or decrease over time. A double moving average requires that you calculate a moving average and then calculate a second moving average using the averages from your first moving average as observations. 6-Step Procedure for Using Moving Averages . When using moving averages, you should use the following 6-step procedure in your analysis: Step 1. Collect the time services data. Step 2. Determine which moving average model to use. • No time-series trend -- use a single moving average. • Time-series trend -- use a double moving average. Step 3. Develop 1-period forecasts using different averaging periods to compare with actual observations to evaluate accuracy. Step 4. Evaluate 1-period forecast accuracy using mean absolute deviations (MADs) between forecasts and actual observations. Step 5. Select the averaging period found to produce the most accurate results. Step 6. Use the moving average in forecasting. 6.1 - Identifying Situations For Use Situations for Use . You can use moving averages in any situation where you are attempting to forecast a variable and you cannot identify or you cannot measure an independent variable except time that appears to be related to changes in the variable. In contract pricing, moving averages are often used to: • Develop estimating rates and factors. For example, most production operations involve substantial amounts of material. When material is used, there is normally some amount of scrap that can no longer be used for its intended purpose. This can include: o Waste from production operations (e.g., sheet metal left over after shapes have been cut from it). o Spoilage (e.g., material that has exceeded its useful shelf life, losses in storage, defective parts, etc.). o Defective parts (e.g., parts that fail inspection during the production process). o Material scrap rates are affected by a variety of factors including production methods, product design, and materials. Because the specific effect of these variables is difficult to identify and measure, scrap rates are commonly estimated using moving averages. o Rates may be calculated in either dollars or units of material and are commonly calculated in one of the following ways: o In calculating such estimating factors, you should track the cost element being estimated (scrap) and the factor base separately over the averaging period. Then you can calculate the moving average rate by completing the rate calculation. However, if that is not possible, you can calculate the moving average using the scrap rates from each data period. The major disadvantage of the latter method is that periods of high and low production receive the same weight in analysis. • Estimate contractor sales volume. A variety of factors affect sales volume (e.g., company products, the economy, Government spending, and many others). Estimates should include current contracts, known future contracts, other known sales, and currently unknown sales. One way of estimating currently unknown sales is with a moving average based on recent sales experience. • Estimate contract requirements . Many contracts obligate the contractor to meet uncertain requirements. A requirements contract for a particular product may require the contractor to meet all Government demand during the contract period. A maintenance contract may require the contractor to respond to unscheduled service calls. In these and similar situations, a moving average can provide estimates of future requirements based on the recent past. • Estimate economic change. A moving average can be used to estimate future economic change based on recent history. For example, wage rates and product price changes (index numbers) can be estimated using moving averages. 6.2 - Determining Which Moving Average Model To Use General Criteria for Model Selection . There are several moving average models that can be used in contract pricing. The two most commonly used are the single moving average and the double moving average. Your decision on which model to use will depend on whether the data indicate a trend (upward or downward) in the values of the dependent variable. If there is: • No time-series data trend - use a single moving average. • Time-series data trend - use a double moving average. Methods to Determine If Data Indicate Dependent Variable Trend . There are three common methods you could use to determine whether or not there is trend in a data set: graphic analysis, regression analysis, and Spearman's rank correlation coefficient. • Graphic Analysis. Graphic analysis entails plotting the data (either manually or by computer) and determining by visual inspection whether or not there is trend in the data. The problem with this technique is that it is not consistently accurate. It is particularly difficult to make a decision when there may or may not be a slight trend. • Regression Analysis. Regression analysis entails the calculation of a least-squares-bestfit (LSBF) estimating equation using time as the independent variable and testing the significance of the slope using the T-test. Though this is an accurate technique, it is rather tedious even when done using a computer. • Spearman's Rank Correlation Coefficient . Spearman's rank correlation coefficient, also known as the Rank Spearman (RANSP) test involves calculation of an RS value and comparing that value with a critical value obtained from a table. This is the test that is most commonly used, because it is accurate and relatively easy to calculate even when done manually. However, before using the RANSP test, assure that the following three conditions have been met: o You must have data from at least four observations. o You must not have reason to suspect a cyclical or seasonal effect. o You must not have reason to suspect that there is a change in trend direction (monotonic trend). Calculating the RANSP . A model to compute the RANSP is provided at RANSP Test Example 1 Assume that you have collected historical quarterly wage data and you want to determine if there is trend in the data. Assume you have the following data in table format in the order in which the values occurred and number the time periods. Quarter (t) Wage Rate (Y) 1 $12.50 2 $11.80 3 $12.85 4 $13.95 5 $13.30 6 $13.95 7 $15.00 8 $16.20 9 $16.10 Using the model at ***, we compute an RS of.9375), which is greater than RS crit of .4667. Thus, we can assume there is a trend in the data. Therefore, you should use a double moving average. RANSP Test Example 2 . Again, assume that you have collected historical quarterly wage data and you want to determine if there is trend in the data. Assume you have the following data in table format in the order in which the values occurred and number the time periods. Quarter (t) Wage Rate (Y) 1 $12.70 2 $12.60 3 $12.00 4 $13.00 5 $12.10 6 $12.50 7 $12.80 8 $13.00 9 $12.85 In this example, the model computes an RS of.4375, which is less than RS crit of .4667. Thus, we would assume that there is no trend in the data. Therefore, you would use a single moving average. 6.3 - Evaluating And Using Single Moving Averages Procedures for Selecting a Single Moving Average for Forecasting . The single moving average is designed to smooth random variation in the estimate. The more data periods you use to calculate a single moving average, the greater the smoothing affect. For example, a 12-period moving average will average out most random variation, because each observation is only one- twelfth of the average. However, a 12-period moving average will be slow to react to a true change in the variable that you are attempting to estimate. On the other hand, a three period moving average will react much faster, because one data point is one-third of the calculation instead of one-twelfth. No averaging period is best for forecasting in all circumstances. You must identify the best averaging period for each situation: Step 1. Develop 1-period forecasts using different available periods so that you can compare forecasts with actual observations to evaluate accuracy. • Use at least three periods of data in developing a moving average. You can calculate 3- period moving averages beginning in Period 3. You can calculate 4-period moving averages beginning in Period 4. For any value of n, you can calculate an n-period single moving average beginning in Period n. • To conduct a meaningful evaluation of forecast accuracy, you must have at least two forecasts and actual data from the same periods for accuracy evaluation. As a result, the largest number of periods (n) that you can use for developing single moving averages is two less than the total number of observations. Step 2. Evaluate 1-period forecast accuracy using mean absolute deviations (MADs) between forecasts and actual observations. Step 3. Select the averaging period found to produce the most accurate results. Calculations Required for Forecast Development . Develop 1-period forecasts using different available periods so that you can compare forecasts with actual observations to evaluate accuracy. Step 1A. Calculate Single Moving Averages . Calculate single moving averages for available averaging periods using the following equation: Where: = A single n-period moving average calculated in Period t Y t = An observation in Period t of the variable being forecast n = The number of time periods in the moving average Step 1B. Develop Forecasts Using Moving Averages. Once you calculate a moving average, you can use that average for forecasting. Where: = A single, n-period, moving average forecast made in Period t for Period t+h n = The number of periods in the moving average t = The period in which the forecast is made h = The horizon, the number of periods you are forecasting into the future Developing 1-Period Forecasts for Example 2 Data. Step 1A. Calculate Single Moving Averages. In the previous section, we determined that we should use a single moving average to forecast future wage rates from the data below. Here we will use the data to demonstrate the procedures for single moving average forecast development. Quarter (t) Wage Rate (Y) 1 $12.70 2 $12.60 3 $12.00 4 $13.00 5 $12.10 6 $12.50 7 $12.80 8 $13.00 9 $12.85 Note that we have observations from nine periods. That means that we can calculate 3-period, 4- period, 5-period, 6-period, and 7-period moving averages. With nine observations, we cannot evaluate forecasts based on a single moving average of more than seven (9 - 2) periods. 3-Period Single Moving Average Quarter (t) Acutal (Y) 3 Y M1 1 $12.70 2 $12.60 3 $12.00 $37.30 $12.43 4 $13.00 $37.60 $12.53 5 $12.10 $37.10 $12.37 6 $12.50 $37.60 $12.53 7 $12.80 $37.40 $12.47 8 $13.00 $38.30 $12.77 9 $12.85 $38.65 $12.88 * Complete terminology for these moving averages is M13,t. To save space the term has been simplified to M1. Step 1B. Develop Forecasts Using Moving Averages : Once we calculate a single moving average, we can use that average to develop a forecast. To evaluate the accuracy of each moving average, we forecast one period into the future so that we can compare the forecast with the actual Y value. For example, the moving average from Period 3, becomes the Forecast for Period 4. 3-Period Single Moving Average Forecast Quarter (t) Actual (Y) 3 Y M1 FM1** 1 $12.70 2 $12.60 3 $12.00 $37.30 $12.43 4 $13.00 $37.60 $12.53 $12.43 5 $12.10 $37.10 $12.37 $12.53 6 $12.50 $37.60 $12.53 $12.37 7 $12.80 $37.40 $12.47 $12.53 8 $13.00 $38.30 $12.77 $12.47 9 $12.85 $38.65 $12.88 $12.77 ** Complete terminology for the forecasts in this column is FM13,t,t+1. To save space the term has been simplified to FM1. We would develop the 4-period, 5-period, 6-period, and 7-period single moving average forecasts using the same procedure. 4-Period Single Moving Average Forecast Quarter (t) Actual (Y) 4 Y M1 FM1 1 $12.70 2 $12.60 3 $12.00 4 $13.00 $50.30 $12.58 5 $12.10 $49.70 $12.43 $12.58 6 $12.50 $49.60 $12.40 $12.43 7 $12.80 $50.40 $12.60 $12.40 8 $13.00 $50.40 $12.60 $12.60 9 $12.85 $51.15 $12.79 $12.60 5-Period Single Moving Average Forecast Quarter (t) Actual (Y) 5 Y M1 FM1 1 $12.70 2 $12.60 3 $12.00 4 $13.00 5 $12.10 $62.40 $12.48 6 $12.50 $62.20 $12.44 $12.48 7 $12.80 $62.40 $12.48 $12.44 8 $13.00 $63.40 $12.68 $12.48 9 $12.85 $63.25 $12.65 $12.68 6-Period Single Moving Average Forecast Quarter (t) Actual (Y) 6 Y M1 FM1 1 $12.70 2 $12.60 3 $12.00 4 $13.00 5 $12.10 6 $12.50 $74.90 $12.48 7 $12.80 $75.00 $12.50 $12.48 8 $13.00 $75.40 $12.57 $12.50 9 $12.85 $76.25 $12.71 $12.57 7-Period Single Moving Average Forecast Quarter (t) Actual (Y) 7 Y M1 F1 1 $12.70 2 $12.60 3 $12.00 4 $13.00 5 $12.10 6 $12.50 7 $12.80 $87.70 $12.53 8 $13.00 $88.00 $12.57 $12.53 9 $12.85 $88.25 $12.61 $12.57 Calculations Required for Evaluating Forecast Accuracy . Evaluate 1-period forecast accuracy using mean absolute deviations (MADs) between forecasts and actual observations. You can use several different statistics to measure the accuracy of a moving average forecast: the range of the error terms, the standard error of the forecast, or the mean absolute deviation of the forecast (MADf). Of these three options, the statistic which best combines the qualities of ease of computation and utility is the MADf. As a result, it is the statistic most commonly used to evaluate moving average accuracy. You can use several different statistics to measure the accuracy of a moving average forecast: the range of the error terms, the standard error of the forecast, or the mean absolute deviation of the forecast (MADf). Of these three options, the statistic which best combines the qualities of ease of computation and utility is the MADf. As a result, it is the statistic most commonly used to evaluate moving average accuracy. The MADf tells us on average how much, in absolute terms, actual values deviated from the forecasted value. Where: MAD F = The mean absolute deviation of the forecast S = Summation of all the variables that follow the symbol |D| = The absolute value of the deviation (i.e., the difference, without regard to sign) between the actual value which occurred and the value forecasted |D| = |Y - F| n = The number of deviations (Ds) computed Evaluate forecast accuracy of single moving averages calculated using the data in Example 2. 3-Period Single Moving Average Evaluation t Actual (Y) M1 FM1 D |D| 1 12.70 2 12.60 3 12.00 12.43 4 13.00 12.53 12.43 0.57 0.57 5 12.10 12.37 12.53 -0.43 0.43 6 12.50 12.53 12.37 0.13 0.13 7 12.80 12.47 12.53 0.27 0.27 8 13.00 12.77 12.47 0.53 0.53 9 12.85 12.88 12.77 0.08 0.08 Total Absolute Deviation 2.01 Mean Absolute Deviation = |D| n = 2.01 6 = 0.34 4-Period Single Moving Average Evaluation t Actual (Y) M1 FM1 D |D| 1 12.70 2 12.60 3 12.00 4 13.00 12.58 5 12.10 12.43 12.58 -0.48 0.48 6 12.50 12.40 12.43 0.07 0.07 7 12.80 12.60 12.40 0.40 0.40 8 13.00 12.60 12.60 0.40 0.40 9 12.85 12.79 12.60 0.25 0.25 Total Absolute Deviation 1.60 Mean Absolute Deviation = |D| n = 1.60 5 = 0.32 5-Period Single Moving Average Evaluation t Actual (Y) M1 FM1 D |D| 1 12.70 2 12.60 3 12.00 4 13.00 5 12.10 12.48 6 12.50 12.44 12.48 0.02 0.02 7 12.80 12.48 12.44 0.36 0.36 8 13.00 12.68 12.48 0.52 0.52 9 12.85 12.65 12.68 0.17 0.17 Total Absolute Deviation 1.07 Mean Absolute Deviation = |D| n = 1.07 4 = 0.27 6-Period Single Moving Average Evaluation t Actual (Y) M1 FM1 D |D| 1 12.70 2 12.60 3 12.00 4 13.00 5 12.10 6 12.50 12.48 7 12.80 12.50 12.48 0.32 0.32 8 13.00 12.57 12.50 0.50 0.50 9 12.85 12.71 12.57 0.28 0.28 Total Absolute Deviation 1.10 Mean Absolute Deviation = |D| n = 1.10 3 = 0.37 7-Period Single Moving Average Evaluation t Actual (Y) M1 FM1 D |D| 1 12.70 2 12.60 3 12.00 4 13.00 5 12.10 6 12.50 7 12.80 12.53 8 13.00 12.57 12.53 0.47 0.47 9 12.85 12.61 12.57 0.28 0.28 Total Absolute Deviation 0.75 Mean Absolute Deviation = |D| n = .75 2 = 0.32 Selecting an Averaging Period . Select the averaging period found to produce the most accurate results. Summary Of MAD Computations n MAD F 3 0.34 4 0.32 5 0.27 6 0.37 7 0.38 The lowest MAD F in this example was attained using a 5-period single moving average. Accordingly, you should select a 5-period single moving average for forecasting. Use the Single Moving Average in Forecasting . Use the moving average with the lowest MAD F for forecasting. Based on an evaluation of the data in Example 2, you should use the most recent 5-period single moving average to forecast for any future period. For example, the forecast for Period 13 would be $12.65. The selection of the most accurate averaging period for forecast development is essential. Different averaging periods can produce substantially different forecasts. For example, using different averaging periods and the data in this example, you could have calculated a wide range of forecasts for Period 13. Period 13 Forecast Comparison n FM1 n,9,13 3 $12.88 4 $12.79 5 $12.65 6 $12.71 7 $12.61 Of these possibilities, the 5-period single moving average forecast, $12.65, appears to be the most reasonable.