BUSI 460 Suggested Answers to Selected Review and Discussion Questions Lesson 7 1. The definitions follow: (a) Time series: Time series data, also known as a data series, consists of observations on a set of variables over time in which the ordering of data is an essential piece of information. Forecasting generally uses time series data, which may have four data patterns known as stationary, trend, seasonal, and cyclical. (b) Autocorrelation: Autocorrelation describes how a variable moves in relation to itself and shows how the current value of a variable moves in relation to the past values of the same variable. Autocorrelation is relevant to forecasting since it allows the analysis of seasonal and other timerelated data patterns. (c) Lag: Lag is a term used in autocorrelation that refers to the interval between the two periods being examined. Lag is relevant in forecasting time series since it examines the relationship of one period's value over the previous period's value during the entire length of the time series. (d) Moving average: A moving average differs from the basic average technique since only a specific number of data points are used to calculate the moving average and it is recalculated each period by dropping the oldest data point and adding a new one. A moving average is useful in forecasting cyclical variables or variables that have a varying pattern. In addition, it is normally used for the smoothing of data that has great variations, effectively eliminating the volatility of a data series. (e) Span: A span is the number of data points used to calculate a moving average. The span is relevant in forecasting since it is the basis of calculating a moving average and as a result, the span must be selected carefully. (f) Smoothing: Exponential smoothing is a forecasting method that is similar to a moving average, however, the forecast applies the greatest weight to the variable s most recent value and places emphasis on recent events. Exponential smoothing is relevant in forecasting since it is useful in the analysis of stationary time series data. Additional methods of exponential smoothing include Holt s method which is useful for forecasting trend data and Winters method which is useful for trend and seasonally adjusted data. 2. Cross-sectional data consists of observations on a given set of variables at one point in time and the order of the observations is not important. Time series data consists of observations on a set of variables that are over time and the ordering of data is essential. Panel data consists of observations on a set of variables that are across many units and also over time. In real estate practice, cross-sectional data on housing, where each house is a variable can be used by a municipal property assessment organization to formulate a model that conducts mass property appraisals. Time series data can be used in real estate practice to analyze the variation in the key interest rate over the past decade. Finally, panel data can be used in real estate to analyze the variation in the lease rates for each of the retail units in a metropolitan area.
460 Review and Discussion Questions: Answer Guide 7 Page 2 3. The four time series patterns are: 1. Stationary: A stationary pattern is evident in the number of properties a Canadian residential appraiser inspects on a weekly basis since the variation in the properties is insignificant and random. 2. Trend: In order to determine the growth of commercial real estate development in Vancouver, the number of development permits approved by the City of Vancouver over the last decade can be analyzed. 3. Seasonal: A seasonal pattern is apparent in Whistler (a popular skiing destination) since the lease rates for residential units peak every winter. 4. Cyclical: The real estate market itself is considered cyclical since it is vulnerable to business cycle risks. As a result, the real estate industry experiences stagnation, recovery, credit-based expansion, booms and crashes. 4. PoolPermits2005 = PoolPermits2004 + (PoolPermits2004 - PoolPermits2003) = 19 + (19 42) = -4 This is a nonsensical result. However, it does highlight a point. Common sense is an integral part of forecasting. 5. Differencing removes trend characteristics from a dataset and effectively transforms a cyclical or seasonal data series into a stationary data series. Differencing helps determine the variation between monthly pricing data by removing the seasonal bias and allowing the examination of each month s average price compared to the previous month in absolute terms. 6. The five forecasting methods and their pros and cons are: i) The naïve forecast assumes that the best predictors of future values are the most recent data available so this forecast estimates the value for a specific time period using the value from the previous period. Pros: This model can provide a short term forecast for an item with limited historical data. Cons: It can only be used to predict one period ahead and if used for more than one period, it assumes that all future periods will be identical to the last period for which data was available. In addition, a random event that occurred in the last period will be repeated every period into the future. The naïve forecast also ignores all past information beyond one period and assumes that the forecast is not affected by the environment. Example: A new concrete forming method has been developed. Analysis of the few developments that have implemented this method is required to determine if it accelerates the time required for the forming process. ii) The average forecast uses the mean of all relevant observations to forecast the next period's value. Pros: It can be easily updated each period by recalculating the average and is useful for data where the process generating the data is stationary. Cons: The average forecast is rarely used in professional forecasting since it is only useful for a variable that is stable and has no trend or other time varying pattern. Example: Utilized when forecasting the number of home inspections a Canadian residential appraiser will conduct on a weekly basis.
460 Review and Discussion Questions: Answer Guide 7 Page 3 iii) The moving average forecast is similar to the basic averaging technique however, it is unique since only a specific number of data points are used to calculate the average. Furthermore, the average is recalculated each period by dropping the oldest data point and adding a new one. Pros: It is useful in forecasting cyclical variables or variables that have a pattern that varies and is also used for the smoothing of data series that are highly volatile. Cons: It is not capable of forecasting seasonal variation and is somewhat weak in forecasting data with a trend because if the trend is consistent, the moving average will underestimate future values. Example: Commonly used by traders as a tool to track and forecast stock prices. iv) The exponential smoothing technique applies the greatest weight to the variable s most recent value and generates a model that will best predict the current level while also using this current level prediction to forecast future levels. Pros: It is useful for data without a trend and better-suited for stationary time series data. Additional exponential techniques such as Holt s method are suited for trend data while Winters method is suited for trend and seasonally adjusted data. Cons: It is extremely complex relative to the other methods. Example: Commonly used to forecast the construction cost index, which allows Project Managers to prepare more accurate pro forma estimates during the pre-development phase. v) The autoregressive integrated moving average forecast combines the three major modelling techniques: Autoregressive regression, Differencing and Moving Average. Pros: This model can handle all data patterns and is more accurate for long term forecasts since it utilizes three different modelling techniques. Cons: Estimating the parameters for this model are extremely complex and it is one of the most complicated methods used to forecast data. As a result, this modeling process can be difficult to explain to clients. Example: Commonly used for long term analysis such as forecasting housing starts in a municipality. 7. The four measures of forecast accuracy are: i) Mean Absolute Deviation (MAD) - this measure is useful when a forecaster wishes to examine the accuracy of different forecasts using the same data series, and is not concerned about whether the average is due to many small errors or only a few large errors ii) Mean Square Error (MSE) - the MSE penalizes large errors. Using this measure, a forecast that predicts a few observations very badly will usually be rejected in favour of a forecast that predicts all observations with only small errors
460 Review and Discussion Questions: Answer Guide 7 Page 4 iii) Mean Absolute Percentage Error (MAPE) - the MAPE may not be extremely useful in practice. It is similar to the MAD, except that each error is divided by the value of the variable itself. Advantages of the MAPE are that it can be used to compare forecasts of different series whose values have different magnitudes and forecasts of one series using different forecasting methods. iv) Mean Percentage Error (MPE) - the mean percentage error (MPE) is used to examine whether a forecast is biased, or whether a forecast is predicting values either consistently high or consistently low. Because the sign of the measure is important, it uses neither absolute values nor squared values. If the mean percentage error is positive, then the forecast is systematically underestimating the value of the observations. We would say that the forecast has a downward bias. On the other hand, if the MPE is negative, then the forecast is systematically overestimating the value of the observations, meaning that the forecast has an upward bias. 8. For research and peer discussion. 9. The forecasts were run using MS-Excel for the second naive forecast and IBM SPSS for the other three models. The procedures for the using SPSS for exponential smoothing and ARIMA can be found with the Online Readings on the Course Resources web-site. (a) The results are inserted following the written answers. (b) the ARIMA choice for the first model was as specified (1, 1, 0)(1, 0, 0) and discussed in the lesson, it produces a flat forecast. The ARIMA parameters of (1, 1, 0)(1, 1, 0) are provided in the table, they cause a wave-like increasing trend similar, if not a little more pronounced than the trend over the past few years. Sequence charts have been provided below. ARIMA(1, 1, 0)(1, 0, 0)
460 Review and Discussion Questions: Answer Guide 7 Page 5 ARIMA(1, 1, 0)(1, 1, 0) (c) high-lighted in blue in the results table (d) the measures of error for the ARIMA forecasts and the exponential smoothing forecast are reasonably close. As such, it would be a toss-up between the two as to which is the best forecast based on those measures. However, forecasting through December 2007 shows the exponential smoothing rising slightly over the period which is more in keeping with expectations (see sequence chart below); although the rising wave of the second ARIMA forecast is also of interest. As such, the exponential smoothing forecast would be the most appropriate for a conservative forecast, the second ARIMA for a more positive forecast. Exponential Smoothing
460 Review and Discussion Questions: Answer Guide 7 Page 6 (e) forecasts for all four forecasting models are shown in the table on the next page. Hold Back Group Second Naive (A) Exponential Smoothing (B) ARIMA (1, 1, 0(1, 0, 0) (C) ARIMA (1, 1, 0)(1, 1, 0) (D) Date Housing Housing Difference Date Measures of Error Measures of Error Measures of Error Measures of Error 07/01/2004 40,185 196 Jul-04 40,185 40,275 40,258 40,767 08/01/2004 42,865 2,680 Aug-04 42,865 40,462 40,372 40,679 09/01/2004 42,718-147 Sep-04 42,718 43,382 43,602 43,398 10/01/2004 42,895 177 Oct-04 42,895 43,169 42,963 43,687 11/01/2004 42,575-320 Nov-04 42,575 43,318 42,928 42,874 12/01/2004 42,148-427 Dec-04 42,148 42,924 42,425 41,248 01/01/2005 40,622-1,526 Jan-05 40,622 42,419 41,653 40,657 02/01/2005 40,018-604 Feb-05 39,096 922 40,714-696 40,239-221 41,026-1,008 03/01/2005 37,525-2,493 Mar-05 37,570-45 40,040-2,515 40,052-2,527 40,485-2,960 04/01/2005 39,950 2,425 Apr-05 36,044 3,906 37,296 2,654 37,044 2,906 37,028 2,922 05/01/2005 40,162 212 May-05 34,518 5,644 39,986 176 40,657-495 41,229-1,067 Jun-05 32,992 40,216 40,243 40,655 Jul-05 31,466 MAD 40,269 MAD 40,283 MAD 41,253 MAD Aug-05 29,940 2,629 40,323 1,510 40,797 1,537 43,205 1,989 Sep-05 28,414 MSE 40,377 MSE 40,769 MSE 43,435 MSE Oct-05 26,888 11,990,920 40,430 3,471,083 40,803 3,781,108 44,245 4,863,559 Nov-05 25,362 MAPE 40,484 MAPE 40,742 MAPE 44,214 MAPE Dec-05 23,836 0.0656 40,537 0.0388 40,660 0.0395 43,844 0.0509 Jan-06 22,310 MPE 40,591 MPE 40,368 MPE 42,060 MPE Feb-06 20,784 0.0650 40,645-0.0034 40,252-0.0031 41,460-0.0144 Mar-06 19,258 40,698 39,774 40,391 Apr-06 17,732 40,752 40,239 42,114 May-06 16,206 40,806 40,280 42,600 Jun-06 14,680 40,859 40,295 43,034 Jul-06 13,154 40,913 40,303 43,457 Aug-06 11,628 40,967 40,401 45,725 Sep-06 10,102 41,020 40,396 45,792 Oct-06 8,576 41,074 40,402 46,327 Nov-06 7,050 41,127 40,391 46,171 Dec-06 5,524 41,181 40,375 45,776 Jan-07 3,998 41,235 40,319 44,104 Feb-07 2,472 41,288 40,297 43,502 Mar-07 946 41,342 40,205 41,816 Apr-07-580 41,396 40,294 43,843 May-07-2,106 41,449 40,302 44,210 Jun-07-3,632 41,503 40,305 44,670 Jul-07-5,158 41,557 40,307 45,169 Aug-07-6,684 41,610 40,325 47,300 Sep-07-8,210 41,664 40,324 47,437 Oct-07-9,736 41,717 40,326 48,092 Nov-07-11,262 41,771 40,323 47,989 Dec-07-12,788 41,825 40,320 47,605