Mobile-Home Shipments

Presented to

Dr. Zerom

Prepared byLing Wang

Szu Tung ChenSarun Sangarungroj

Spring 2015

Table of Contents

Executive Summary...................................................................................................................2

1. The Forecasting Problem...............................................................................................3

2. Examination of Data Patterns........................................................................................3

3. Analysis of Seasonality..................................................................................................5

4. Analysis of Trend...........................................................................................................8

5. Analysis of Cycle.........................................................................................................10

6. Analysis of Fitted Forecasts.........................................................................................11

7. Forecasts......................................................................................................................12

8. Evaluation of Forecast Accuracy.................................................................................13

9. Conclusion...................................................................................................................14

Appendix 1...............................................................................................................................15

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Executive Summary

Kim Brite and Larry Short, developers of a series of exclusive mobile-home parks, are considering about opening more mobile-home sites. They want to manage their cash flow more efficiently, so they hired us to forecast mobile-home shipments (MHS), because MHS appears to influence the vacancy rates and the rate at which they can fill newly opened parks.

In this report, we use multiplicative classical time series decomposition as a forecasting method. First, we examine the data patterns. The data we have to analyze is the quarterly rate of mobile-home Shipments from 1988 Q1 to 2003 Q4. After applying autocorrelation analysis and differencing, we find that the time series has a clear quarterly seasonality.

Next, we use ratio-to-moving average (RMA) method to decompose the time series into three individual components. Here we discover that the seasonal indexes of Q2 and Q3 are above average while the indexes of Q1 and Q4 are below average. We also find out that the trend actually exists, which is slightly decreasing over time from 1988 to 2003. Most importantly, we discover a cycle pattern, which significantly affected the fluctuation of time series of MHS. Then, we use our multiplicative decomposition forecasting method to do a fitted forecast. The results are quite good since our forecasts follow the actual historical data very well.

Lastly, we forecast all four quarters of year 2004 using Box-Jenkins and our decomposition methods. After we obtain the actual data of year 2004, we calculate the MAPE of both methods and the better method is multiplicative decomposition with a MAPE of 39.36% while the Box-Jenkins has a MAPE of 120%.

In conclusion, Kim Brite and Larry Short should follow the forecasts of the multiplicative decomposition method where they should prepare for more shipments in Q2 and Q3 and less in Q1 and Q4. This report will show all the steps mentioned above in details.

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1. The Forecasting Problem Kim Brite and Larry Short have developed a series of exclusive mobile-home parks. Considering to expand their business and open more such parks, they need to measure their cash flow. mobile-home shipments (MHS) is the key factor related to the cash flow management. Therefore, they need a better forecast of MHS based on 16 years data from 1988 Q1 to 2003 Q4.

Since Kim and Larry’s business is a typical small business and not complicated, we decide to use multiplicative classical time series decomposition to forecast the sales for 4 quarters of 2004. The detailed forecasting report is showed as below.

2. Examination of Data Patterns

Exhibit 2-1Quarterly Rate of Mobile Home Shipment from 1988 Q1 to 2003 Q4

We are now analyzing the data of quarterly rate of mobile-home shipments from 1988 Q1 to 2003 Q4. By visualizing the time series data in Exhibit 3-1, we can see a seasonal pattern existed because the rate of mobile-home shipments are usually higher during the period of Q2 and Q3 than the period of Q1 and Q4. However, we are not able to find any clear trend existed in this time series data by visualization.

To further examine the data pattern, we apply Autocorrelation Analysis to the time series data in Exhibit 2-2.

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Exhibit 2-2Autocorrelation Function

Exhibit 2-2 shows the Autocorrelation Function calculated by Forecast Pro XE. From the autocorrelation function, we still cannot say there is a seasonal pattern or trend existed in this time series data. Though there are spikes in lag 3, lag 6, and lag 9, the time series data acts irregularly after lag 9.

To further clearly examine the trend and seasonal pattern in the time series, we use the differencing method to remove the effect of one pattern and verify if another pattern exists. We first examine the existence of a seasonality, and we take the first order simple differencing to remove possible trend effects from the autocorrelation function. This is shown in Exhibit 2-3.

Exhibit 2-3Autocorrelation Function with Simple Differencing

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From the above correlogram, we can clearly see significant positive spikes and negative spikes every 4 lags, which indicates a clear quarterly seasonality. Next, we take the first order differencing to remove possible seasonal effects from the autocorrelation function to examine the existence of a trend. This is shown in Exhibit 2-4.

Exhibit 2-4Autocorrelation Function with Seasonal Differencing

From the above correlogram, we see a declined trend from lag 1 to lag 26 and then a slightly increased trend after lag 26. However, it is hard to say that the trend is going to increase because this trend might be affected by a cycle.

As a result, by using autocorrelation analysis to explore the time series data, we can conclude that there is a significant quarterly seasonal pattern and a trend that might with a cycle for quarterly rate of mobile-home shipments from 1988 Q1 to 2003 Q4.

3. Analysis of Seasonality

We apply ratio-to-moving average (RMA) method to decompose the time series into three individual components: seasonality, trend, cycle.

First, we decompose seasonal factors by four steps.

Step 1: Finding Centered Moving Average (CMA)

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On the first step, we remove the short-term fluctuations from seasonal pattern so that the longer-term trend and cycle components can be more clearly identified. Seasonal pattern can be removed by calculating an appropriate moving average (MA) for the time series. The MA should contain the same number of periods as there are in the seasonality. Thus, due to the quarterly data, a four-period MA is appropriate.

We use the equation MA = (Yt-2 + Yt-1 + Yt + Yt+1) / 4; the result is in Appendix1.

Because we have four quarters, which is an even number of periods, to make the moving average represents a center value, we have to use centered moving average (CMA): CMAt = (MAt+ MAt+1) / 2. The result is showed in Exhibit 3-1.

Exhibit 3-1 De-seasonalized data-- CMA

Exhibit 3-2Comparing De-seasonalized Value with Actual Value

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We compare the de-seasonalized value with actual value in Exhibit 3-2. The data appears smoother after the seasonal variations have been removed.

Step 2: Seasonal Factor (SF)

The seasonal variations have been removed by moving average method, so the value of CMA represents the de-seasonalized data. By comparing the actual value with the de-seasonalized value, we get the seasonality factor (SF): SFt = Yt / CMAt. The result is in Appendix1.

1 is the average SF. When SF is bigger than 1, it means a period in which Y is bigger than the yearly average. When SF is smaller than 1, it means a period in which Y is smaller than the yearly average. For example, the SF in the 1st quarter in 1989 is 0.89761, it means the value in quarter 1 is smaller than average in 1989.

Step 3: Seasonal Index (SI)

Since the SF for each period are bound to have some variability, we calculate a seasonal index (SI) for each season. The SI for a season is simply the average of seasonal factors of that season. The result is in Table 3-1.

Table 3-1The seasonal Indexes and the Sum

Q1 Q2 Q3 Q4 SUM

0.90348 1.09039 1.06796 0.93464 3.99646

The sum of seasonal indexes is 3.9932, which smaller than 4, so we have to adjust the seasonal indexes and the result is Table 3-2.

Table 3-2Adjusted Seasonal Indexes

Q1 Q2 Q3 Q4

0.90428 1.09135 1.0689 0.93547

To better visualizing the seasonal indexes and CMA value, we use column chart (Exhibit 3-3) to display the adjusted indexes and line chart to display the CMA value as follow:

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Exhibit 3-3The Adjusted Seasonal Indexes of Sales

After round the adjusted SI, the seasonal indexes for MHS are as follows: First quarter: 0.90 Second quarter: 1.10 Third quarter: 1.07 Fourth quarter: 0.94

These SIs add to 4 as expected. Based on the value of SI, the strongest seasons for mobile-home parks are spring (quarter 2) and summer (quarter 3). And the business in winter (quarter 1 and quarter 2) shows a weak performance than average.

4. Analysis of Trend

The long-term trend is estimated based on the deseasonalized data. To determine the trend, we use a simple linear equation CMA= a+ b(TIME) to generate an estimate of the trend value of the centered moving average for the historical and forecast periods. This new series is centered moving average trend (CMAT).

By simply running a linear regression of CMA on time, the equation of CMAT is: CMAT=62.55898-0.0531(TIME)The result is in Exhibit 3-4.

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Exhibit 3-4The Trend -- CMAT

We can see from Exhibit 3-4, the trend exists. The value slightly decreases from 1988 to 2003.

Exhibit 3-5Comparing MHS, CMA, and CMAT

We can compare the actual value, de-seasonalized value, and trend in Fugure 3-5.

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5. Analysis of Cycle

The cyclical component of a time series is the extended wavelike movement about the long-term trend. It is measured by a cycle factor (CF), which is the ratio of the centered moving average (CMA) to the centered moving average trend (CMAT). That is:CF=CMA/CMAT

The result is in Exhibit 3-6. A cycle greater than 1 represents that the de-seasonalized value in that period is above the long-term trend of the data. When CF is less than 1, it means that the de-seasonalized value in that period is below the long-term trend of the data.

Exhibit 3-6Cycle

We can see from Exhibit 3-6 that from 1988 Q1 to 1990 Q3 and 1994 Q2 to 2001, the de-seasonalized value were below the trend. While from 1990 Q4 to 1994 Q1 the de-seasonalized value were above the trend. We can see that the CF in Exhibit 3-6 moves above and below the line at 1.00 exactly as the centered moving average moves above and below the trend line in Exhibit 3-5. By isolating the cyclical factor in Exhibit 3-6, we can better analyze its movements over time.

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6. Analysis of Fitted Forecasts

Exhibit 6-1 shows how well our time series decomposition model follows the historical pattern in the data.

Exhibit 6-1Comparison between Decomposition Model and Historical Data

Because we use multiplicative forecasting method, we simply multiply seasonal indexes (SI), trend (CMAT), and cyclical factor (CF) to generate forecasts value in the fitted period, which is showed in Exhibit 6-1.

As you can see in exhibit 6-1, our decomposition model clearly forecasts the higher mobile-home shipments (MHS) during Q2 and Q3 in every year. In addition, it also forecasts the decreased trend from 1988 to 2003 of mobile-home shipments (MHS), along with the cycle of mobile-home shipments (MHS).

Therefore, we can say that our decomposition model goes pretty well along with the historical data. This decomposition model follows not only the seasonal pattern but also the trend and the cycle of the historical fitted data.

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7. Forecasts

First, we use the Box-Jenkins models to forecast the cycle forecast for 2004. The result shown in Exhibit 7-1.

Exhibit 7-1Forecasts of Mobile Home Shipments for 4 Quarters of 2004 by Box-Jenkins Models

TIME CMAT CF SI Forecast

2004-Q1 65 59.107 1.529 0.90 81.33

2004-Q2 66 59.054 1.568 1.09 100.93

2004-Q3 67 59.001 1.606 1.07 101.39

2004-Q4 68 58.948 1.645 0.94 91.15

Next, we are going to use dynamic regression to estimate a linear relationship between the cyclical factor and the interest rate. To do so, we must find the values of the interest rate for 2004 first. Since we do not have the true values of the interest rate for 2004, we use Forecast Pro’s Expert Selection to forecast them. The results are shown in the table below.

Exhibit 7-2Forecasts of Interest Rate for 2004 by ForecastPro’s Expert Selection

Forecasted IR

2004-Q1

8.76

2004-Q2

8.79

2004-Q3

8.83

2004-Q4

8.87

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Then, we use ForecastPro’s dynamic regression to forecast cyclical factors for 2004 while using Interest Rate as a predictor variable. The results are shown in Exhibit 7-3 and Exhibit 7-4.

Exhibit 7-3Forecasts of Cyclical Factors for 4 Quarters of 2004 by Dynamic Regression

After we obtain all the values, we can calculate the forecasts of mobile-home shipments. The results are shown in Exhibit 7-4.

Exhibit 7-4Forecasts of Mobile Home Shipments for 4 Quarters of 2004

TIME CMAT CF SI Forecast

2004-Q1 65 59.107 1.0034 0.90 53.38

2004-Q2 66 59.054 1.0033 1.09 64.58

2004-Q3 67 59.001 1.0033 1.07 63.34

2004-Q4 68 58.948 1.0032 0.94 55.59

8. Evaluation of Forecast Accuracy

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To measure the forecast accuracy, we only use MAPE to determine whether the model is good or not. From the previous part, we got the forecasts from both Box-Jenkins method and the Decomposition method. We can calculate MAPE for both methods easily by comparing them to the actual values. The results are shown in Exhibit 8-1.

Exhibit 8-1Actual Sales Compared to the Forecasts of Box-Jenkins and Decomposition Methods of the Year 2004 and MAPE of Both Methods

Actual ValueBox-Jenkins Decomposition

2004-Q1 35.4 81.33 53.38

2004-Q2 47.3 100.93 64.58

2004-Q3 47.2 101.39 63.34

2004-Q4 40.9 91.15 55.59

MAPE 1.201996 0.39356

The MAPE for the unknown future quarters (2004 Q1 - 2004 Q4) we got from Box-Jenkins model is very large at 120%, while the MAPE we got from the Decomposition method is around 39.36%. Therefore, the Decomposition method, with the interest rate as a predictor for forecasting CF, is better than Box-Jenkins due to a lower MAPE.

For Box-Jenkins method, the cyclical factor (CF) we got for the forecast period assumed that the cycle goes up, which means it thinks the sales will go up. However, the actual sales for mobile-home shipments didn’t increase. The MAPE of 120% is pretty high and we think this inaccuracy might mainly result by the difficulty of forecasting the cyclical factor (CF) for the forecast period.

9. Conclusion

From our analysis, there is a clear quarterly seasonality in the time series with a strong cycle pattern and small decreasing trend from 1988 to 2003. Based on the seasonal indexes and the actual data, Kim and Larry should prepare their shipments a lot more for Q2 and Q3 since the number of shipments tends to go above average during these two quarters.

On the other hand, since the sales are likely to go below average for Q1 and Q4, Kim and Larry should create some promotions such as giving discounts or coupons to stimulate the sales.

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Interest Rate is also an important factor since it improves the accuracy of forecasted Cyclical Factors which leads to a better overall accuracy of the forecasts of MHS. Ultimately, when they make decisions for their business, they should consider our forecasts data which is based on the multiplicative decomposition method, as it provides forecasts with a better MAPE of 39.36% than a Box-Jenkins method with 120% MAPE.

Appendix 1

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