BUSI 344 LESSON 8 SUPPLEMENT TIME ADJUSTMENT ILLUSTRATION

BUSI 344 LESSON 8 SUPPLEMENT TIME ADJUSTMENT ILLUSTRATION The "Ontario" database used in Lesson 8 did not have sufficient market movement to require a time adjustment. However, because this is a common requirement in modelling, we will illustrate the process here using different data that does require a time adjustment. If your price data needs time adjusting and you do not account for this, you risk having the model adjust for the influence of time within the coefficient values. This is particularly problematic in an explanatory model, where you are looking for coefficients that accurately explain each variable's contribution to value. To illustrate time adjustments, we have provided another database called "prgeorge". This provides sales data from Prince George, BC, from 1 Jan 2001 to 31 Oct 2001. Because our modelling exercise is for assessment purposes, the valuation date is 1 July 2001. We also have data on the assessed values as of July 1, 2000 (TOTALVAL). We will determine if the market has moved by comparing sale prices to the 2000 assessed values, to see if there is a pattern in sale prices over time. We will create a new variable for the sale to assessment ratio (SAR), which is PRICE / TOTALVAL. The SAR shows how close sale prices are to assessments. If SAR = 1, this means sale price was equal to assessed value; SAR less than one means sale price was below assessed value; and SAR greater than 1 means sale price was above assessed value. We will examine the SAR by MONTH using a scatterplot with a regression line and R-square value fitted to the result. 2.0 1.8 1.6 1.4 1.2 1.0 SAR.8 Rsq = 0.0981 0 2 4 6 8 10 12 MONTH The regression line indicates a small, but continuous increase in SAR over the 10 months. In other words, sales prices exhibited a small, steady increase from January to October 2001. However, the data are irregular and do not show a constant increase from month to month. Also, the correlation, as indicated by R-square, is also quite low. This indicates that the use of a constant percentage change per month is not appropriate and an adjustment for each individual month may be more reasonable. As the R-square value is very low, a test should be carried out to determine if the variations in the average monthly values of the SAR variable are significant enough to warrant a time adjustment. This can be accomplished with a Kruskal-Wallis test. The expected mean rank in each month should be approximately equal to the middle value in the database; in this case there are 453 observations so the expected value of the mean is approximately 226. If the observed mean rank for each month varies significantly from 226, this would indicate that the months are not valued equally and that there is some form of difference in sale price in various months. 1

Test Statistics a, b SAR BUSI 344 Lesson 8 Supplement Analyze ý Nonparametric Tests ý K independent samples Enter SAR as Test Variable and Month as Group Variable. Click Define Range button, enter 1 as Minimum value and 10 as maximum value for Month. Continue ý OK Kruskal-Wallis Test MONTH N Mean Rank SAR 1 14 93.25 2 21 158.43 3 24 178.46 4 44 152.02 5 70 191.10 6 78 252.76 7 67 258.16 8 64 267.32 9 41 271.33 10 30 286.82 Total 453 Chi-Square 67.222 df 9 Asymp. Sig..000 a) Kruskal Wallis Test b) Grouping Variable: MONTH Only the sixth and seventh months have mean ranks close to the expected value of 226. Months one to five have lower than expected values and months eight to ten have higher than expected values. The chi-square statistic for SAR is 67.222 and the probability of obtaining a test statistic of this amount if the months are equally valued is 0. This means that a time adjustment is required. To determine the amount of time adjustment needed, go to Analyze ý Compare Means ý Means and calculate the average sale/assessment ratio (SAR) by MONTH. SAR is the Dependent Variable and MONTH is the Independent Variable. SAR MONTH Mean N Std. Deviation 1.00 103.3720 14 8.3847 2.00 109.0207 21 9.7523 3.00 112.4092 24 12.6029 4.00 108.8555 44 11.0203 5.00 112.6485 70 15.6912 6.00 117.9241 78 11.9597 7.00 118.3222 67 12.8425 8.00 118.8127 64 11.1832 9.00 121.1895 41 16.6506 10.00 120.9177 30 12.1378 Total 115.7516 453 13.5816 2

Time Adjustment Illustration This report provides the mean ratio of sale price to assessed value for the months of January to October of 2001. For example, in January, sale prices were an average of 3.37% higher than July 2000 assessed values, while in October sale prices were almost 21% higher than assessed values. This is illustrated in Figure 1. Figure 1 Sale to Assessment Ratio by Month We want to adjust for variation in sales prices during 2001 so that we have a value for each property that represents the estimated sale price if the property had been sold on July 1 of 2001. We will find the average SAR for July 1 and then adjust the sales prices in other months so that the SAR in these months is equal to that found on July 1. Monthly averages tend to create a value for approximately the 15th of the month and our valuation date is July 1, so it is necessary to average the results of June and July to get the July 1 average. (117.92 + 118.32) 2 = 118.12, rounded to 118.1 The actual monthly time adjustment factors must be calculated manually and are shown in the table below: 3

BUSI 344 Lesson 8 Supplement MONTH AVERAGE CALCULATION FACTOR 1 (Jan) 103.4 118.1 103.4 1.142 2 (Feb) 109 118.1 109.0 1.083 3 (Mar) 112.4 118.1 112.4 1.051 4 (April) 108.9 118.1 108.9 1.084 5 (May) 112.6 118.1 112.6 1.049 6 (June) 117.9 118.1 117.9 1.002 7 (July) 118.3 118.1 118.3 0.998 8 (Aug) 118.8 118.1 118.8 0.994 9 (Sept) 121.2 118.1 121.2 0.974 10 (Oct) 120.9 118.1 120.9 0.977 Open a syntax file and create the variable TAFACTOR, the time adjustment factor, as follows: COMPUTE TAFACTOR=MONTH. RECODE TAFACTOR (1=1.142)(2=1.083)(3=1.051)(4=1.084)(5=1.049)(6=1.002)(7=.998) (8=.994)(9=.974) (10=.977). This factor will adjust the sales price of properties in each month in order to achieve a desired average SAR of 118.1 for each month. For example, in January the mean SAR was only 103.4. To adjust this SAR up to 118.1, all sales prices in July must be multiplied by 1.142, or increased by 14.2%. Similarly, the average SAR in October is 120.9; this requires all sales prices in October to be multiplied by 0.977, or a.3% downward adjustment. This is illustrated in Figure 2. Figure 2 Adjusting Monthly Sale-to-Assessment Ratios If you had repeat sales, as we did in the "Ontario" database, you could use these pairs to test the accuracy of your calculated time adjustments. 4

Time Adjustment Illustration Go to the syntax file and create the variable TASPRICE as follows: COMPUTE TASPRICE=TAFACTOR*PRICE. COMPUTE TA_SAR=TASPRICE/TOTALVAL. time adj. sale price = time adj. factor price time adjused SAR We can confirm the time adjustment has been effective by viewing the scatterplot of the time-adjusted SAR by month. There is no longer an upward sloping line and the R-square is close to 0. An advantage of this time adjustment method is that each month was calculated and adjusted separately, accounting for possible fluctuations during the year or for seasonal variation. This is an improvement on single property adjustments, where the norm is a simple linear adjustment applied evenly, e.g., 1% per month. NOTE: If your valuation date was at the end of the period, e.g., October 31, 2001, then you would instead use 120.9 as the numerator in the factor calculations. October's factor would be 1.0 and all but September would be above 1.0, illustrating the need to increase their sale price to account for the rising market between their sale date and the valuation date. Alternative Method: Using Price Per Square Foot If you need to test for a time adjustment and do not have the luxury of the previous assessment in your database, then you can only compare the sales across the months to see if there is a pattern. For example, you could view the scatterplot of Price against Month to see the pattern. However, if you control for size, this may help isolate the effect of time, particularly if your database is small and there are insufficient sales in each month to get a good average. To control for size, you may use either price per square foot of living area or price per the combined total square footage of living area and lot size. The premise here is that an increase or decrease in the price per square foot of living area (or living area and lotsize) over the time period should indicate a rising or falling market. There is an assumption here that your data is fairly homogeneous. 5

BUSI 344 Lesson 8 Supplement You would then use similar procedures to the foregoing: examine a scatterplot between price per square foot variable and the sale month, compare means, and run the Kruskal-Wallis tests. If a time adjustment was necessary, the same methodology would be applied to create factors for Price. Alternative Method: Including Time Variables in the Regression Equation An alternative methodology to account for the need for a time adjustment is to use regression analysis. For our example, we can create binary variables for each month in the sale date range, meaning that a sale in January 2001 will have the Jan variable = 1 and 0 for all other month variables. In our regression model, we then include the month variables with the other independent variables. For example, we will run a regression with Price as the dependent variable and independent variables flarea1, flrarea2, bsmtfin (finished basement area), baseste (basement suite), bedrooms, baths, familyrm, fireplcs, garagefl and carptfl (yes/no for garage or carport), and 9 of the month variables. We will omit July as our reference variable, so that all the other months are calculated in comparison to July. We find the following results. Model Summary(b) Adjusted R Std. Error of Model R R Square Square the Estimate 1.880(a).775.765 11,732.953 a Predictors: (Constant), OCT, carptfl, lot_size, FEB, baseste, JAN, MAR, SEP, flrarea2, APR, AUG, fireplcs, bedrooms, MAY, garagefl, familyrm, JUN, flrarea1, baths, bsmtfin b Dependent Variable: price Coefficients(a) Unstandardized Coefficients Standardized Coefficients Model B Std. Error Beta t Sig. 1 (Constant) 2013.801 4400.864.458.647 bedrooms -956.738 1089.381 -.023 -.878.380 baths 2242.703 1709.850.047 1.312.190 garagefl 6421.661 1431.595.122 4.486.000 carptfl 2276.421 1316.963.046 1.729.085 baseste -3108.523 2722.116 -.032-1.142.254 familyrm 2740.316 1424.454.065 1.924.055 fireplcs 3488.318 926.574.113 3.765.000 flrarea1 56.161 4.051.460 13.864.000 flrarea2 42.812 3.322.367 12.888.000 bsmtfin 10.049 2.842.138 3.536.000 lot_size.999.318.079 3.140.002 JAN -8603.111 3475.387 -.062-2.475.014 FEB -8982.000 2972.081 -.078-3.022.003 MAR -5275.248 2827.606 -.049-1.866.063 APR -7404.938 2289.203 -.091-3.235.001 MAY -1645.308 2050.435 -.025 -.802.423 JUN 876.956 1969.923.014.445.656 AUG 3573.594 2071.089.052 1.725.085 SEP 902.217 2357.192.011.383.702 OCT 3478.286 2613.126.036 1.331.184 a Dependent Variable: price 6

Time Adjustment Illustration The coefficients for the month variables indicate the relative effect of time on sale price in this model. For example, sales in January 2001 have a coefficient of -$8,603, meaning sales in January need to be increased by $8,603 on average. In contrast, sales in October are adjusted by +$3,478, meaning sales in October need be adjusted downward by $3,478 on average. These coefficients account for market movement between January and October 2001, in comparison to a July 1 valuation date. The low t-values and high Sig for several of the months, particularly those around the valuation date, call their accuracy into question. More investigation should be undertaken on these suspect months as to whether an adjustment is warranted. For example, you could use stepwise regression to identify the months that need a significant time adjustment in comparison to the July target month. However, we will not carry this investigation here. Viewing the scatterplot of the new SAR using time-adjusted values against month shows virtually no relationship between SAR and month of sale. A possible drawback to this method is that our coefficients indicate dollar adjustments by month, rather than the traditional monthly percentage adjustments for time normally found in appraisal. Our dollar adjustments are less-than-ideal in appraisal terms, because high-priced properties and low-priced properties are receiving the same dollar adjustment and therefore very different percentage adjustments. Ideally, the time adjustment should be multiplicative, but we cannot have multiplicative adjustments in our additive model. However, we might indirectly bring a multiplicative adjustment into the model by multiplying each month binary by a size variable, such as lot size, and then using these nine "lot size by month" variables instead of the month binaries. 7