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Financial Forecasting – Time Series Models Essay

The economic recession felt in the United States since the collapse of the housing market in 2007 can be seen by various trends in the housing market. This collapse claimed some of the largest financial institutions in the U.S. such as Bear Sterns and Lehman Brothers, as they held over-leveraged positions in the mortgage backed securities market. Credit became widely available to unqualified borrowers during the nineties and the early part of the next decade which caused bankers to act predatorily in their lending practices, as they could easily sell and package subprime mortgage loans on leverage. This act caused a bubble that would later burst when unqualified homebuyers began defaulting on their loans causing a tremendous downfall in the U.S housing market. Understanding which direction key market factors, such as the housing market, are going can help re-establish stability in the market, which is at an all-time premium. This paper is designed to help better predict the direction of the housing market in the future via the use of time series models, in an effort to re-establish a sense of stability in the housing market.

The following chart (Figure 1) represents the time series data for non-seasonally adjusted home sales in the U.S. (NHS) from January 1975 to February 2012. The length of this period is significant because over a long period of time we can analyze trend, seasonality, cycles, and irregularity allowing us to better understand the future direction of the market. Trend is the long term change in the level of data. We can find trend in the data by simply looking at the chart and observing the general direction of the data over a long period of time. These trends can be deduced to a consistent change in the mean level of the data over a significant period of time, keeping in mind that seasonality will occur year over year therefore annual recurring changes to the level of data should not account for an increase or decrease in the trend mean. Seasonality is the regular fluctuations in levels of data in a time series that occur every year at the same time of the year. Seasonality is often seen in data that fluctuates regularly in accordance with calendar seasons.

When analyzing this data we must also take into account cycles. Cyclical data can be recognized by it smooth elongated upward and downward movements on a long term scale. These reactions are more irregular than seasonal patterns, but more regular than a change in the trend. Generally the cause of a cycle is less apparent right away and occurs because of the ups and downs in the economy making it harder to predict. In Figure 1 we can observe a distinct upward trend until late in 2005, with strong seasonality, and three distinct cycles. The final component that we must acknowledge is irregularity. Irregularities are the random fluctuations that are not affected by the other three components making it the hardest to predict or rationalize. There is some irregularity in Figure 1, but it does not seem to be dominant as most of the fluctuations noticed in the time series could be rationalized by one of the previous three factors.

Data source: National Association of Realtors

One way to verify a trend in a time series is to analyze a k-period plot of autocorrelations, also known as an autocorrelation function (ACF). If a trend is present we should notice a gradual decline, however if we see a steep decline we should note that there is no trend. In Figure 2, which represents time series data for non-seasonally adjusted home sales in the U.S. (NHS) from January 1975 to February 2012 we can observe a gradual decline meaning a positive trend is present. Additionally we can use the rule of thumb stating each value in the 12-period plot of ACF is greater than .0947 (2/446) and thus greater than the upper limit, representing statistical significance from zero and concluding that a trend is present.

Now that we have verified the presense of a trend in the data we will look to verify the seasonality we saw earlier represented by regularly reoccurring fluctuations in the levels of data in accordance with the calendar seasons. To do this we will use an autocorrelation function for the first differenced new home sales data. We will use a larger sample, in this case 24 months, so that we can see the regularly reoccurring fluctuations from one year to the next. When we look at the graph in Figure 3 we notice great increases with lag 12 and lag 24. The jumps seen in lags 12 and 24 confirms the presense of seasonality as they are above the upper limit representing statistical significance. F

Time Series and Regression Models for New One-Family Houses Sold Since the NHS data has been shown to have trend and seasonality we will evaluate the data using four different time series models and compare the results of each to see which model is the most accurate. The models we are going to use are the Modified Naïve model, Winters Exponential Smoothing model, Time Series Decomposition, and Autoregressive Integrated Moving Average (ARIMA). We will also test a multiple regression model to attempt to forecast future NHS, while taking into consideration independent variables. Regression models determine the future direction of the dependent variable based on the forecasts of the independent variable(s). Often times this can lead to a less accurate forecast as too much emphasis is being applied to the correlation of the independent variables to the dependent variable.

In reality large ranges of macroeconomic data such as NHS vary because of numerous variables that may not be taken into account. The Multiple Regression model in Table 1 will have NHS as the dependent variable and use the 30-year conventional mortgage and the Seasonally Adjusted Disposable Person Income as the independent variables. This data comes from the Federal Reserve Bank of St. Louis. We are going to use two separate periods in our analysis. The first period that we are going to use is our historical data from January 1975 through August 2011. The last six months of model from September 2011 through February 2012 is our hold out model in which we test the forecasted NHS results against the actual NHS during the same span of time to test the accuracy of the models forecast’s.

We will use ForecastX software to run the models in an attempt to determine which model is the most accurate and thus should be used to forecast NHS to obtain the clearest picture of the future direction of the market. The two error measurements we will use to determine accuracy are mean absolute percentage error (MAPE) and root mean square error (RMSE). To obtain MAPE, we first divide the forecast error or error (actual value – forecast value) by the actual value to yield the percentage error and then calculate the mean of the absolute percentage errors. For RMSE, we first square the forecast error and then take the squared root of the mean of the squared errors.

Multiple regression model| 29.54%| 19.07| 32.07%| 105.58%| 43.12| 176.00%| *The mean for the historical period is 59.46 and the holdout period mean is 24.5 We can deduce the most accurate forecasting model from Table 1’s forecasting error results. The model with the least amount of error is the most accurate, which in this case is the Time Series Decomposition model with exponential smoothing for the historical period.

However the ARIMA model contains the least amount of error for the holdout period which would leads me to believe that it would be the most accurate indicator over the six month time period from March 2012 to August 2012, however when the model is run over the entire range of data to forecast the next six month period the upper and lower limits drastically trend away from each other each month resulting in a very vague forecast with little confidence. Therefore we will use the Time Series Decomposition to forecast the ex-ante forecast from March 2012 to August 2012.

As I touched on earlier, the housing market is a key indicator as to the overall health of the market. Like other markets, the housing market is a victim to cyclical fluctuations. However, if investors and market participants can accurately forecast cyclical fluctuations in the housing market or in interest rates they will have more confidence in the market and be able to make more aggressive moves, spurring economic activity. The forecast summary in Figure 4 shows a continuation in the current NHS cycle through July 2012. This information could be used by a real estate investor to gain insight as to the right time to buy. If NHS continues to decline then we can assume that prices will continue to fall with it, and an investor can wait for the right time to buy real estate at the lowest price.

Conclusion

The current outlook for NHS for one-family homes in the U.S. according to Figure 4, is seemingly positive. We have experienced a great recession since 2007, and in that recession we have reached depths unrealized since 1975 for NHS. This is shocking because of the population inflation in the country throughout that time, and is really telling of the severity of the recession we endured. Although, with that being said I think that the depleted NHS statistics will encourage investors to jump in and start buying up new property, as well as family that might have never been financially able to do it before. With these low levels in sales we have seen some tremendous deals becoming available, and at the same time banks are starting to lend again. Considering all of those factors I think that NHS will again start to trend positively.

References

[1] “Total New Houses Sold: Thousands.” Economagic: Economic Time Series Page. N.p., n.d. Web. 25 Apr. 2012. <http://www.economagic.com/em-cgi/data.exe/cenc25/stagemon01>. [2] Wilson, J. Holton, and Barry Keating. Business forecasting: with forecastX. 6. ed. Boston: McGraw-Hill/Irwin, 2009. Print. [3] Forecast X 7.1. John Galt Solutions, Inc.


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