Financial Forecasting - Time Series Models

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Time Series Models for Forecasting New One-Family Houses Sold in the United States Introduction
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 Data Pattern of New One-Family Houses Sold in the U.S.

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. Figure 1

New One-Family Houses Sold (NHS), in thousands, in the U.S. January 1975 to February 2012

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...
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