Real Estate Modelling and Forecasting by Chris Brooks and Sotiris Tsolacos_3 doc

An overview of regression analysis 75 x y 100 80 60 40 20 0 10 20 30 40 50 Figure 4.1 Scatter plot of two variables, y and x to get the line that best ‘fits’ the data. The researcher would then be seeking to find the values of the parameters or coefficients, α and β, that would place the line as close as possible to all the data points taken together. This equation (y = α + βx) is an exact one, however. Assuming that this equation is appropriate, if the values of α and β had been calculated, then, given a value of x, it would be possible to determine with certainty what the value of y would be. Imagine – a model that says with complete certainty what the value of one variable will be given any value of the other. Clearly this model is not realistic. Statistically, it would correspond to the case in which the model fitted the data perfectly – that is, all the data points lay exactly on a straight line. To make the model more realistic, a random disturbance term, denoted by u, is added to the equation, thus: y t = α + βx t + u t (4.2) where the subscript t (= 1, 2, 3, ) denotes the observation number. The disturbance term can capture a number of features (see box 4.2). Box 4.2 Reasons for the inclusion of the disturbance term ● Even in the general case when there is more than one explanatory variable, some determinants of y t will always in practice be omitted from the model. This might, for example, arise because the number of influences on y is too large to place in a single model, or because some determinants of y are unobservable or not measurable. ● There may be error s in the way that y is measured that cannot be modelled. 76 Real Estate Modelling and Forecasting ● There are bound to be random outside influences on y that, again, cannot be modelled. For example, natural disasters could affect real estate performance in a way that cannot be captured in a model and cannot be forecast reliably. Similarly, many researchers would argue that human behaviour has an inherent randomness and unpredictability! How, then, are the appropriate values of α and β determined? α and β are chosen so that the (vertical) distances from the data points to the fitted lines are minimised (so that the line fits the data as closely as possible). The parameters are thus chosen to minimise collectively the (vertical) distances from the data points to the fitted line. This could be done by ‘eyeballing’ the data and, for each set of variables y and x, one could form a scatter plot and draw on a line that looks as if it fits the data well by hand, as in figure 4.2. Notethatitisthevertical distances that are usually minimised, rather than the horizontal distances or those taken perpendicular to the line. This arises as a result of the assumption that x is fixed in repeated samples, so that the problem becomes one of determining the appropriate model for y given (or conditional upon) the observed values of x. This procedure may be acceptable if only indicative results are required, but of course this method, as well as being tedious, is likely to be impre- cise. The most common method used to fit a line to the data is known as ordinary least squares (OLS). This approach forms the workhorse of econo- metric model estimation, and is discussed in detail in this and subsequent chapters. x y Figure 4.2 Scatter plot of two variables with a line of best fit chosen by eye An overview of regression analysis 77 x y 10 8 6 4 2 0 01234567 Figure 4.3 Method of OLS fittingalinetothe data by minimising the sum of squared residuals Two alternative estimation methods (for determining the appropriate val- ues of the coefficients α and β) are the method of moments and the method of maximum likelihood. A generalised version of the method of moments, due to Hansen (1982), is popular, although the method of maximum likeli- hood is also widely employed. 1 Suppose now, for ease of exposition, that the sample of data contains only five observations. The method of OLS entails taking each vertical distance from the point to the line, squaring it and then minimising the total sum of the areas of squares (hence ‘least squares’), as shown in figure 4.3. This can be viewed as equivalent to minimising the sum of the areas of the squares drawn from the points to the line. Tightening up the notation, let y t denote the actual data point for obser- vation t, ˆ y t denote the fitted value from the regression line (in other words, for the given value of x of this observation t, ˆ y t is the value for y which the model would have predicted; note that a hat [ˆ] over a variable or parameter is used to denote a value estimated by a model) and ˆ u t denote the residual, which is the difference between the actual value of y and the value fitted by the model – i.e. (y t − ˆ y t ). This is shown for just one observation t in figure 4.4. What is done is to minimise the sum of the ˆ u 2 t . The reason that the sum of the squared distances is minimised rather than, for example, finding the sum of ˆ u t that is as close to zero as possible is that, in the latter case, some points will lie above the line while others lie below it. Then, when the sum to be made as close to zero as possible is formed, the points above the line would count as positive values, while those below would count as negatives. These distances will therefore in large part cancel each other out, which would mean that one could fit virtually any line to the data, so long as the sum of the distances of the points above the line and the sum of the distances of the points below the line were the same. In that case, there would not be 1 Both methods are beyond the scope of this book, but see Brooks (2008, ch. 8) for a detailed discussion of the latter. 78 Real Estate Modelling and Forecasting x y û t y t x t y t ˆ Figure 4.4 Plot of a single observation, together with the line of best fit, the residual and the fitted value a unique solution for the estimated coefficients. In fact, any fitted line that goes through the mean of the observations (i.e. ¯ x, ¯ y) would set the sum of the ˆ u t to zero. On the other hand, taking the squared distances ensures that all deviations that enter the calculation are positive and therefore do not cancel out. Minimising the sum of squared distances is given by minimising ( ˆ u 2 1 + ˆ u 2 2 + ˆ u 2 3 + ˆ u 2 4 + ˆ u 2 5 ), or minimising  5  t=1 ˆ u 2 t  This sum is known as the residual sum of squares (RSS) or the sum of squared residuals. What is ˆ u t , though? Again, it is the difference between the actual point and the line, y t − ˆ y t . So minimising  t ˆ u 2 t is equivalent to minimising  t (y t − ˆ y t ) 2 . Letting ˆα and ˆ β denote the values of α and β selected by minimising the RSS, respectively, the equation for the fitted line is given by ˆ y t = ˆα + ˆ βx t . Now let L denote the RSS, which is also known as a loss function.Takethe summation over all the observations – i.e. from t = 1 to T ,whereT is the number of observations: L = T  t=1 (y t − ˆ y t ) 2 = T  t=1 (y t − ˆα − ˆ βx t ) 2 (4.3) L is minimised with respect to (w.r.t.) ˆα and ˆ β, to find the values of α and β that minimise the residual sum of squares to give the line that is closest An overview of regression analysis 79 to the data. So L is differentiated w.r.t. ˆα and ˆ β, setting the first derivatives to zero. A derivation of the ordinary least squares estimator is given in the appendix to this chapter. The coefficient estimators for the slope and the intercept are given by ˆ β =  x t y t − T ¯ x ¯ y  x 2 t − T ¯ x 2 (4.4) ˆα = ¯ y − ˆ β ¯ x (4.5) Equations (4.4) and (4.5) state that, given only the sets of observations x t and y t , it is always possible to calculate the values of the two parameters, ˆα and ˆ β, that best fit the set of data. To reiterate, this method of finding the optimum is known as OLS. It is also worth noting that it is obvious from the equation for ˆα that the regression line will go through the mean of the observations – i.e. that the point ( ¯ x, ¯ y) lies on the regression line. 4.5 Some further terminology 4.5.1 The data-generating process, the population regression function and the sample regression function The population regression function (PRF) is a description of the model that is thought to be generating the actual data and it represents the true relationship between the variables. The population regression function is also known as the data-generating process (DGP). The PRF embodies the true values of α and β, and is expressed as y t = α + βx t + u t (4.6) Note that there is a disturbance term in this equation, so that, even if one had at one’s disposal the entire population of observations on x and y,it would still in general not be possible to obtain a perfect fit of the line to the data. In some textbooks, a distinction is drawn between the PRF (the underlying true relationship between y and x) and the DGP (the process describing the way that the actual observations on y come about), but, in this book, the two terms are used synonymously. The sample regression function (SRF) is the relationship that has been estimated using the sample observations, and is often written as ˆ y t = ˆα + ˆ βx t (4.7) Notice that there is no error or residual term in (4.7); all this equation states is that, given a particular value of x, multiplying it by ˆ β and adding ˆα will 80 Real Estate Modelling and Forecasting give the model fitted or expected value for y, denoted ˆ y. It is also possible to write y t = ˆα + ˆ βx t + ˆ u t (4.8) Equation (4.8) splits the observed value of y into two components: the fitted value from the model, and a residual term. The SRF is used to infer likely values of the PRF. That is, the estimates ˆα and ˆ β are constructed, for the sample of data at hand, but what is really of interest is the true relationship between x and y – in other words, the PRF is what is really wanted, but all that is ever available is the SRF! What can be done, however, is to say how likely it is, given the figures calculated for ˆα and ˆ β, that the corresponding population parameters take on certain values. 4.5.2 Estimator or estimate? Estimators are the formulae used to calculate the coefficients – for example, the expressions given in (4.4) and (4.5) above, while the estimates, on the other hand, are the actual numerical values for the coefficients that are obtained from the sample. Example 4.1 This example uses office rent and employment data of annual frequency. These are national series for the United Kingdom and they are expressed as growth rates – that is, the year-on-year (yoy) percentage change. The rent series is expressed in real terms – that is, the impact of inflation has been extracted. The sample period starts in 1979 and the end value is for 2005, giving twenty-seven annual observations. The national office data provide an ‘average’ picture in the growth of real rents in the United Kingdom. It is expected that regions and individual markets have performed around this growth path. The source of the rent series is constructed by the authors using UK office rent series from a number of real estate consultancies. The employment series is that for finance and business services published by the Office for National Statistics (ONS). Assume that the analyst has some intuition that employment (in partic- ular, employment growth) drives growth in real office rents. After all, in the existing literature, employment series (service sector employment or financial and business services employment) receive empirical support as a direct or indirect driver of office rents (see Giussani, Hsia and Tsolacos, 1993, D’Arcy, McGough and Tsolacos, 1997, and Hendershott, MacGregor and White, 2002). Employment in business and finance is a proxy for business conditions among firms occupying office space and their demand for office An overview of regression analysis 81 25 (yoy%) (yoy%) 20 15 10 5 −5 −10 −1 −2 −3 −15 −20 −25 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 0 (a) Real office rents (b) Employment in financial and business services (EFBS) 8 7 6 5 4 3 2 1 0 Figure 4.5 Plot of the two variables space. Stronger employment growth will increase demand for office space and put upward pressure on rents. The relationship between economic drivers and rents is not as simple, however. Other influences can be impor- tant – for example, how quickly the vacancy rate adjusts to changes in the demand for office space, and, in turn, how rents respond to changing vacancy levels; how much more intensively firms utilise their space and what spare accommodation capacity they have; whether firms can afford a higher rent; and so forth. Nonetheless, a lack of good-quality data (for exam- ple, national office vacancy data in the United Kingdom) can necessitate the direct study of economic series and rents, as we discuss further in chapter 6. A starting point to study the relationship between employment and real rent growth is a process of familiarisation with the path of the series through time (and possibly an examination of their statistical properties, although we do not do so in this example), and the two series are plotted in figure 4.5. The growth rate of office rents fluctuated between nearly −25 per cent and 20 per cent during the sample period. This magnitude of variation in the growth rate is attributable to the severe cycle of the late 1980/early 1990s in the United Kingdom that also characterised office markets in other countries. The amplitude of the rent cycle in more recent years has lessened. Employment growth in financial and business services has been mostly positive in the United Kingdom, the exception being three years (1981, 1991 and 1992) when it was negative. The UK economy experienced a prolonged recession in the early 1990s. We observe greater volatility in employment growth in the early part of the sample than later. Panels (a) and (b) of figure 4.5 indicate that the two series have a general tendency to move together over time so that they follow roughly the same cyclical pattern. The scatter plot of employment and real rent growth, shown in figure 4.6, reveals a positive relationship that conforms with our expectations. This positive 82 Real Estate Modelling and Forecasting 8 6 4 2 0 0 Real rents (yoy %) Employment in FBS (yoy %) 10 20 30 −2 −4 −30 −20 −10 Figure 4.6 Scatter plot of rent and employment growth relationship is also confirmed if we calculate the correlation coefficient, which is 0.72. The population regression function in our example is RRg t = α + βEFBSg t + u t (4.9) where RRg t is the growth in real rents at time t and EFBSg t is the growth in employment in financial and business services at time t. Equation (4.9) embodies the true values of α and β, and u t is the disturbance term. Esti- mating equation (4.9) over the sample period 1979 to 2005, we obtain the sample regression equation R ˆ Rg t = ˆα + ˆ βEFBSg t =−9.62 +3.27EFBSg t (4.10) The coefficients ˆα and ˆ β are computed based on the formulae (4.4) and (4.5) – that is, ˆ β =  x t y t − T ¯ x ¯ y  x 2 t − T ¯ x 2 = 415.64 − 6.55 363.60 − 238.37 = 3.27 and ˆα = 0.08 −3.27 × 2.97 =−9.62 The sign of the coefficient estimate for β (3.27) is positive. When employ- ment growth is positive, real rent growth is also expected to be positive. If we examine the data, however, we observe periods of positive employment growth associated with negative real rent growth (e.g. 1980, 1993, 1994, 2004). Such inconsistencies describe a minority of data points in the sam- ple, otherwise the sign on the employment coefficient would not have been positive. Thus it is worth noting that the regression estimate indicates that the relationship will be positive on average (loosely speaking, ‘most of the time’), but not necessarily positive during every period. An overview of regression analysis 83 x y 0 Figure 4.7 No observations closetothey-axis The coefficient estimate of 3.27 is interpreted as saying that, if employ- ment growth changes by one percentage point (from, say, 1.4 per cent to 2.4 per cent – i.e. employment growth accelerates by one percentage point), real rent growth will tend to change by 3.27 percentage points (from, say, 2 per cent to 5.27 per cent). The computed value of 3.27 per cent is an aver- age estimate over the sample period. In reality, when employment increases by 1 per cent, real rent growth will increase by over 3.27 per cent in some periods but less than 3.27 per cent in others. This is because all the other factors that affect rent growth do not remain constant from one period to the next. It is important to remember that, in our model, real rent growth depends on employment growth but also on the error term u t , which embod- ies other influences on rents. The intercept term implies that employment growth of zero will tend on average to result in a fall in real rent growth by 9.62 per cent. A word of caution is in order, however, concerning the reliability of estimates of the coefficient on the constant term. Although the strict inter- pretation of the intercept is indeed as stated above, in practice it is often the case that there are no values of x (employment growth, in our example) close to zero in the sample. In such instances, estimates of the value of the intercept will be unreliable. For example, consider figure 4.7, which demonstrates a situation in which no points are close to the y-axis. In such cases, one could not expect to obtain robust estimates of the value of y when x is zero, as all the information in the sample pertains to the case in which x is considerably larger than zero. 84 Real Estate Modelling and Forecasting 20 (yoy %) Actual Fitted (%) 15 10 5 0 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 −5 −10 −15 −20 (a) Actual and fitted values for RR (b) Residuals −25 20 15 10 5 0 −5 −10 −15 −20 −25 Figure 4.8 Actual and fitted values and residuals for RR regression Similar caution should be exercised when producing predictions for y using values of x that are a long way outside the range of values in the sample. In example 4.1, employment growth takes values between −1.98 per cent and 6.74 per cent, only twice taking a value over 6 per cent. As a result, it would not be advisable to use this model to determine real rent growth if employment were to shrink by 4 per cent, for instance, or to increase by 8 per cent. On the basis of the coefficient estimates of equation (4.10), we can generate the fitted values and examine how successfully the model replicates the actual real rent growth series. We calculate the fitted values for real rent growth as follows: R ˆ Rg 79 =−9.62 +3.27 × EFBSg 79 =−9.62 +3.27 × 3.85 = 2.96 R ˆ Rg 80 =−9.62 +3.27 × EFBSg 80 =−9.62 +3.27 × 3.15 = 0.68 . . . . . . . . . (4.11) R ˆ Rg 05 =−9.62 +3.27 × EFBSg 05 =−9.62 +3.27 × 2.08 =−2.83 The plot of the actual and fitted values is given in panel (a) of figure 4.8. This figure also plots, in panel (b), the residuals – that is, the difference between the actual and fitted values. The fitted values series replicates most of the important features of the actual values series. In particular years we observe a larger divergence – a finding that should be expected, as the environment (economic, real estate market) within which the relationship between rent growth and employ- ment growth is studied, is changing. The difference between the actual and fitted values produces the estimated residuals. The properties of the residu- als are of great significance in evaluating a model. Key misspecification tests are performed on these residuals. We study the properties of the residuals in detail in the following two chapters. [...]... this case 92 Figure 4.11 Effect on the standard errors of xt2 large Real Estate Modelling and Forecasting y 0 Figure 4.12 Effect on the standard errors of xt2 small x y 0 x (4) The term xt2 affects only the intercept standard error and not the slope standard error The reason is that xt2 measures how far the points are away from the y-axis Consider figures 4.11 and 4.12 In figure 4.11, all the points are... of the form yt = α + β + ut xt (4.15) the regression can be estimated using OLS by setting zt = 1 xt and regressing y on a constant and z Clearly, then, a surprisingly varied array of models can be estimated using OLS by making suitable 86 Real Estate Modelling and Forecasting transformations to the variables On the other hand, some models are intrinsically non-linear – e.g γ yt = α + βxt + ut (4.16)... hold, and the sample size is sufficiently large The issue of non-normality, how to test for it, and its consequences is discussed further in chapter 6 ˆ Standard normal variables can be constructed from α and β by subtracting ˆ the mean and dividing by the square root of the variance: α−α ˆ ∼ N(0, 1) √ var(α) and ˆ β −β ∼ N(0, 1) √ var(β) The square roots of the coefficient variances are the standard... thus a large-sample, or asymptotic, property The assumptions that E(xt ut ) = 0 and var(ut ) = σ 2 < ∞ are sufficient to derive the consistency of the OLS estimator 88 Real Estate Modelling and Forecasting 4.8.2 Unbiasedness ˆ The least squares estimates of α and β are unbiased That is, ˆ E(α) = α ˆ (4.18) ˆ E(β) = β (4.19) and Thus, on average, the estimated values for the coefficients will be equal to... which 90 Real Estate Modelling and Forecasting ˆ is ut , is used: s2 = 1 T ˆt u2 (4.25) This estimator is a biased estimator of σ 2 , though An unbiased estimator of s 2 is given by s = 2 ˆt u2 T −2 (4.26) ˆt where u2 is the residual sum of squares, so that the quantity of relevance for the standard error formulae is the square root of (4.26): s= ˆt u2 T −2 (4.27) s is also known as the standard error... principles using some algebra, and this is left to the appendix to this chapter Some general intuition is now given as to why the formulae for the standard errors given by (4.20) and (4.21) contain the terms that they do and in the form that they do The presentation offered in box 4.4 loosely follows that of Hill, Griffiths and Judge (1997), which is very clear Box 4.4 Standard error estimators (1) The... value 4.9 Precision and standard errors ˆ Any set of regression estimates α and β are specific to the sample used in ˆ their estimation In other words, if a different sample of data was selected from within the population, the data points (the xt and yt ) will be different, leading to different values of the OLS estimates ˆ Recall that the OLS estimators (α and β) are given by (4.4) and (4.5) It ˆ would... estimate for the standard error of the regression (s) to calcuˆ late the standard error of the estimators α and β For the calculation of ˆ 2 2 ˆ SE(β), we have s = 6.97, EFBSt = 363.60, T × EFBS = 238.37, and thereˆ fore SE(β) = 0.62 and SE(α) = 2.29 ˆ With the standard errors calculated, the results for equation (4.10) are written as ˆ R Rgt = −9.62 + 3.27EFBSgt (2.29) (0.62) (4.28) The standard error estimates... parameters (its mean and variance) This makes the algebra involved in statistical inference considerably simpler than it otherwise would have 94 Real Estate Modelling and Forecasting been Since yt depends partially on ut , it can be stated that, if ut is normally distributed, yt will also be normally distributed Further, since the least squares estimators are linear combinations of ˆ the random variables... α and β determined by OLS ˆ will have a number of desirable properties; such an estimator is known as a best linear unbiased estimator (BLUE) What does this acronym represent? ˆ ˆ ● ‘Estimator’ means that α and β are estimators of the true value of α and β ˆ ˆ ● ‘Linear’ means that α and β are linear estimators, meaning that the forˆ are linear combinations of the random variables (in mulae for α and . zero. 84 Real Estate Modelling and Forecasting 20 (yoy %) Actual Fitted (%) 15 10 5 0 1979 1981 19 83 1985 1987 1989 1991 19 93 1995 1997 1999 2001 20 03 2005 1979 1981 19 83 1985 1987 1989 1991 19 93 1995 1997 1999 2001 20 03 2005 −5 −10 −15 −20 (a). (yoy%) 20 15 10 5 −5 −10 −1 −2 3 −15 −20 −25 1979 1981 19 83 1985 1987 1989 1991 19 93 1995 1997 1999 2001 20 03 2005 1979 1981 19 83 1985 1987 1989 1991 19 93 1995 1997 1999 2001 20 03 2005 0 (a) Real office rents (b). +3. 27EFBSg t (4.10) The coefficients ˆα and ˆ β are computed based on the formulae (4.4) and (4.5) – that is, ˆ β =  x t y t − T ¯ x ¯ y  x 2 t − T ¯ x 2 = 415.64 − 6.55 36 3.60 − 238 .37 = 3. 27 and ˆα