Forecasting Project
House Sales in USA
Presented by:
Michelle Deets
Ravin Seju
Ankan Sinha

March 7, 2016
Professor Dawit Zerom
ISDS 526

Executive Summary
The data in the report “Monthly total houses sold in the United States for the period January
1978 through July 2007” is time series data representing sales in thousands of units. The data has not been seasonally adjusted. Our project was to produce forecasts of housing sales by creating a model using Forecast Pro’s Expert Selection Method. The model was generated by withholding 2 years of data and creating a forecast based upon the data from January 1978 to July 2005. We provided fit measures based upon MAPE and RMSE and evaluated the model’s accuracy MAPE,
RMSE, and GMRAE from Forecast Pro’s out-of-sample statistic evaluation table. The MAPE numbers show that the forecast expands from a 7.95% error at the beginning of the holdout period and quickly grows to 32% error within 24 months. The acceptability of this error depends upon which managers are using it. The housing industry touches many fields, from moving to painting to construction to land purchases. This large of an error might could be unacceptable given the amount of risk and resources involved in construction of new single family homes; a manager might prefer to only use the first 6 months of forecasted data to stay within a 10% error range. Looking at the results of the forecast’s graph, the actual data (Exhibit A, represented by the green line) shows that actual housing sales dropped from July 2005 to July 2007. However, our forecasting model predicted that housing sales would vacillate but reach steady seasonal peaks
(represented by the red line). Using this forecast could devastate a company if it invested with the thought that the houses would sell at predictable intervals. This holdout analysis shows that we cannot be 100% certain about what the future holds.
Exhibit A: Holdout Analysis

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Section I
Exhibit 1-1: Monthly House Sales from 1978 to 2007 with Forecasts

Date
2005-Aug
2005-Sep
2005-Oct
2005-Nov
2005-Dec
2006-Jan
2006-Feb
2006-Mar
2006-Apr
2006-May
2006-Jun
2006-Jul
2006-Aug
2006-Sep
2006-Oct
2006-Nov
2006-Dec
2007-Jan
2007-Feb
2007-Mar
2007-Apr
2007-May
2007-Jun
2007-Jul

MAPE
7.95%
7.64%
5.51%
7.13%
6.79%
7.87%
11.18%
12.43%
13.42%
14.22%
14.93%
16.95%
18.33%
19.36%
20.98%
21.90%
22.33%
24.02%
26.42%
28.28%
29.13%
30.30%
31.39%
32.41%

To check if our model works, we must check the fitted values for explanatory power. Here, we can see the fitted MAPE for our expert selection model. It shows that on average we have a 6.34% error in the fitted portion of the data. This is pretty low and so it should suffice to explain the values in the holdout analysis. Next, we check the Root
Mean Square Error (RMSE) and we can see the fitted values are only off by 4.59 thousand houses. This is very low considering the data seems to be around 20 thousand at the lowest period. So, overall the model fits the historical data very well.
Now, we check in the holdout period for accuracy. We can see that the
MAPE in the holdout analysis is 32.4% at 2 years out. This is because the further out in the future one predicts the more prone to error the model becomes. We can see the MAPE increase in the table below. For the RMSE, we note that we are off by 29.46 thousand units sold at 2 years out. This is a very high amount of error. The Geometric Mean
Relative Absolute Error (GMRAE) shows that relative to the naïve model, the Exponential smoothing: No trend, Multiplicative seasonality model has only 75% of the error. So, this model is actually better than the naïve model.

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Fit Measure: MAPE 6.34%
Fit Measure: RMSE

4.59

Accuracy: MAPE

32.41%

Accuracy: RMSE

29.46

Accuracy: GMRAE

A manager in the housing sector may use these forecasts to decide on time consuming and expensive plans to meet future demand for housing. Because houses cannot be built with a short lead times, it is even more important to get accurate measurements. A manager might not accept such a MAPE of 32.40% at 2 years forecast.

0.75

In other industries, lead times can often be reduced to take advantage of more accurate forecasts. Just-In-Time philosophy aims to reduce lead time by making suppliers be on standby at short notice effectively handing risk to them for overstocking. This is often used by retailers to shorten lead time and therefore increase accuracy.

Section II
A) Data for monthly total house sales was given for the time period between January 1987 and July 2007 and is graphically displayed in Exhibit 1.
Exhibit 2-1: Monthly House Sales from 1978 to 2007

By visually inspecting the data, it’s difficult to identify a trend. The time series is completely unexpected with many highs and lows. The time series is not stationary because the chart starts with a downward trend which dips down all-the-way till the end of the year 1982. It gradually peaks up around 1986.
Again, we notice decreasing sales that reaches its lowest nearing the end of year 1990. The trend picks up again and steadily moves upward from 1995 to 2005. The seasonality seems to be additive because the magnitude of spikes do not increase or decrease proportionally by a specific number. We also see possible cyclical behavior displayed in one year cycles. For example, from 1978 to the end of 1982, the sales peaks at the middle of each year around the period of March to June then sales plunges at the end of the year which again picks up the next following year. This pattern looks to reflect cyclical or there is some influence that has led to increase in houses sold every year. We use autocorrelation to give us a more scientific approach to prove if a trend or a seasonality exists.

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B) Autocorrelations are a way to measure the correlation of a time series with itself. It measures if values at time lags move together. High autocorrelation values means there is a lot of variance explained by the movement of one time period in another time period. This is a good way to check for trend and seasonality, since in both these cases lagged time periods move together, either positively or inversely.
We are checking the ACF to find out if there is seasonality or trend in the dataset. One would expect a sharp drop in the auto correlation function if there is no trend and also lack of peaks at multiples of certain time lags if there is no seasonality. If we check the Autocorrelation graph in Exhibit 2-2, we can see this is not the case.
Exhibit 2-2: Autocorrelation graph

The Correlogram seems to decrease very linearly with occasional spikes every multiple of
12 time lags. This pattern persists all the way up to period 17. This suggests a trend with particularly high values of autocorrelation at multiples of period 12, suggesting a seasonal component. To see both the trend and seasonality we will do simple and seasonal differencing. Please note that the black dotted band is the 95% confidence level for our results, so values higher than this band are statistically significant at p=.05. Simple Differencing is the removal of the trend: i.e., the difference in actual values of the trend at time lag 1. (see Exhibit 2-3)

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Exhibit 2-3: Autocorrelation graph with simple differencing only

This simple differencing removed the trend and shows that the data still has autocorrelation at multiples of period 12. Generally, seasonality occurs for monthly data at 12 periods apart. The statistically significant auto-correlations at multiples of 12 show that there is in fact some seasonality in the data every 12 months.
Now let’s look at seasonal differencing. Seasonal Differencing is the removal of the difference of y values at lag 12 for monthly data. (see Exhibit 2-4)
Exhibit 2-4: Autocorrelation graph with seasonal differencing only

One can see that the autocorrelations for the lagged values are significantly different than 0 at 95% interval. This means that there is a trend present and lagged values are moving together positively at lags up to 9 periods apart. The autocorrelations go down linearly indicating a trend.
Looking at the data in general, it seems the seasonal peaks occur every March. This coincides with general research that house buyers generally prefer spring to buy houses since they receive their tax refunds during this time. It should also be noted that the increase in the seasonality seems to be
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additive rather than multiplicative. This means that the seasonality goes up by some set amount rather than a proportionally to the level of the seasonality.
If we do first order differencing on both simple and seasonal differencing, we get the following
Correlogram. (see Exhibit 2-5)
Exhibit 2-5: Autocorrelation function with first order simple and seasonal differencing

We can see that without trend and seasonality, this time series looks like randomness, or white noise.
White noise has autocorrelations that hover around 0 for almost all time lags.
Also, one can see some signs of cyclical behavior where peaks occur initially at 8 years or so from 1978 to 1986, but this is unpredictable as with most cyclical activity.

C) The time series is not stationary because the time series chart starts with a downward trend, then briefly peaks around 1986, then steadily trends upward from 1995 to 2005.
Trend and seasonality will affect the value of the time series at different times and force us to choose a certain time for observation. We can create a stationary time series to level the mean. Since this data set is also cyclical, it helps to have a stationary view of the data.
To make our housing sales time series stationary, we set the simple differencing to 1 to subtract each month from the prior, thereby removing the difference between this month and last month, and leaving only the horizontal movement (see Exhibit 2-3). We left the seasonal differencing at 0, so we did not subtract 12 lagged values.
The graph (see Exhibit 2-5) shows the effect of using both simple and seasonal differencing; it leaves us with only random noise, which is why we chose the previous option of only simple differencing (see
Exhibit 2-3).

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Exhibit 2-3: Autocorrelation graph

Exhibit 2-3 shows a consistent ACF going downward till lag 15. This means values are positively correlated with each other up to this time lag. As lag increases, the less the data correlates to previous data. Exhibit 2-3: Autocorrelation graph with simple differencing only

Exhibit 2-3 shows the data adjusted to become stationary. We have removed the trend shown in the original data. The seasonal correlations are still evident at multiples of lag 12. These are statistically significant at 95% and so are not random.

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Exhibit 2-6: Autocorrelation function with trend and seasonality removed

In Exhibit 2-6, trend and seasonality are both removed, and the graph is only left with white noise and randomness. Only the periods at 12 and 24 are statistically significant at 95%.
Exhibit 2-4: Autocorrelation function with seasonal differencing only

Exhibit 2-4 looks at seasonal differencing with values at lag 12 for monthly data. The autocorrelations for the lagged values are significantly different than 0 at 95% interval. The trend appears to move inversely, from a positive to a negative correlation, around period 36, but the significance level is so low that we cannot be sure that this is an accurate correlation.

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D) The types of models that should be used in forecasting need to consider if the data are trended, seasonal, or highly irregular or volatile. Significant cost savings correspond to a few percentage points reduction in forecast errors, so the model must be focused on reducing error. The model also needs to be conservative with its estimates.
Given that our data has seasonality, trend, and level, our models would have to include these components. Only Winter’s Exponential Smoothing, time series decomposition and Box-Jenkins can account for both trending and seasonality. Box-Jenkins uses a somewhat abstract concept of autocorrelation, while exponential smoothing is built upon clear cut features like trend, seasonality, and level, making exponential the more preferred method in the business world. It should be noted that our trend seems to be slightly volatile as it goes up and down and up again. Exponential deals with volatility better than Box-Jenkins. There also appears to be cyclical patterns overall. Regression and Box-Jenkins would do well in data that is highly cyclical as housing sales data would be. These models would be our recommendation on how to deal with this housing data. Forecast Pro may not always be correct and it should be up to the user to use their own expertise to judge which model is best with the aid of Forecast
Pro.

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