Property Estate Modelling and Forecasting_1 doc

Real estate analysis: statistical tools 43 3.1.3 Panel data Panel data have the dimensions of both time series and cross-sections – e.g. the monthly prices of a number of REITs in the United Kingdom, France and the Netherlands over two years. The estimation of panel regressions is an interesting and developing area, but will not be considered further in this text. Interested readers are directed to chapter 10 of Brooks (2008) and the references therein. Fortunately, virtually all the standard techniques and analysis in econo- metrics are equally valid for time series and cross-sectional data. This book concentrates mainly on time series data and applications, however, since these are more prevalent in real estate. For time series data, it is usual to denote the individual observation numbers using the index t and the total number of observations available for analysis by T. For cross-sectional data, the individual observation numbers are indicated using the index i and the total number of observations available for analysis by N. Note that there is, in contrast to the time series case, no natural ordering of the observations in a cross-sectional sample. For example, the observations i might be on city office yields at a particular point in time, ordered alphabetically by city name. So, in the case of cross-sectional data, there is unlikely to be any useful information contained in the fact that Los Angeles follows London in a sample of city yields, since it is purely by chance that their names both begin with the letter ‘L’. On the other hand, in a time series context, the ordering of the data is relevant as the data are usually ordered chronolog- ically. In this book, where the context is not specific to only one type of data or the other, the two types of notation (i and N or t and T ) are used interchangeably. 3.1.4 Continuous and discrete data As well as classifying data as being of the time series or cross-sectional type, we can also distinguish them as being either continuous or discrete, exactly as their labels would suggest. Continuous data can take on any value and are not confined to take specific numbers; their values are limited only by precision. For example, the initial yield on a real estate asset could be 6.2 per cent, 6.24 per cent, or 6.238 per cent, and so on. On the other hand, discrete data can take on only certain values, which are usually integers 1 (whole numbers), and are often defined to be count numbers – for instance, the number of people working in offices, or the number of industrial units 1 Discretely measured data do not necessarily have to be integers. For example, until they became ‘decimalised’, many financial asset prices were quoted to the nearest 1/16th or 1/32nd of a dollar. 44 Real Estate Modelling and Forecasting transacted in the last quarter. In these cases, having 2,013.5 workers or 6.7 units traded would not make sense. 3.1.5 Cardinal, ordinal and nominal numbers Another way in which we can classify numbers is according to whether they are cardinal, ordinal or nominal. This distinction is drawn in box 3.2. Box 3.2 Cardinal, ordinal and nominal numbers ● Cardinal numbers are those for which the actual numerical values that a particular variable takes have meaning, and for which there is an equal distance between the numerical values. ● On the other hand, ordinal numbers can be interpreted only as providing a position or an ordering. Thus, for cardinal numbers, a figure of twelve implies a measure that is ‘twice as good’ as a figure of six. Examples of cardinal numbers would be the price of a REIT or of a building, and the number of houses in a street. On the other hand, for an ordinal scale, a figure of twelve may be viewed as ‘better’ than a figure of six, but could not be considered twice as good. Examples include the ranking of global office markets that real estate research firms may produce. Based on measures of liquidity, transparency, risk and other factors, a score is produced. Usually, in this scoring, an office centre ranking second in transparency cannot be said to be twice as transparent as the office market that ranks fourth. ● The final type of data that can be encountered would be when there is no natural ordering of the values at all, so a figure of twelve is simply different from that of a figure of six, but could not be considered to be better or worse in any sense. Such data often arise when numerical values are arbitrarily assigned, such as telephone numbers or when codings are assigned to qualitative data (e.g., when describing the use of space, ‘1’ might be used to denote offices, ‘2’ to denote retail and ‘3’ to denote industrial, and so on). Sometimes, such variables are called nominal variables. ● Cardinal, ordinal and nominal variables may require different modelling approaches or, at least, different treatments. 3.2 Descriptive statistics When analysing a series containing many observations, it is useful to be able to describe the most important characteristics of the series using a small number of summary measures. This section discusses the quantities that are most commonly used to describe real estate and other series, which are known as summary statistics or descriptive statistics. Descriptive statistics are calculated from a sample of data rather than being assigned on the basis of theory. Before describing the most important summary statistics used in Real estate analysis: statistical tools 45 work with real estate data, we define the terms population and sample, which have precise meanings in statistics. 3.2.1 The population and the sample The population is the total collection of all objects to be studied. For example, in the context of determining the relationship between risk and return for UK REITs, the population of interest would be all time series observations on all REIT stocks traded on the London Stock Exchange (LSE). The population may be either finite or infinite, while a sample is a selec- tion of just some items from the population. A population is finite if it contains a fixed number of elements. In general, either all the observations for the entire population will not be available, or they may be so many in number that it is infeasible to work with them, in which case a sample of data is taken for analysis. The sample is usually random, and it should be representative of the population of interest. A random sample is one in which each individ- ual item in the population is equally likely to be drawn. A stratified sample is obtained when the population is split into layers or strata and the num- ber of observations in each layer of the sample is set to try to match the corresponding number of elements in those layers of the population. The size of the sample is the number of observations that are available, or that the researcher decides to use, in estimating the parameters of the model. 3.2.2 Measures of central tendency The average value of a series is sometimes known as its measure of location or measure of central tendency. The average value is usually thought to measure the ‘typical’ value of a series. There are a number of methods that can be used for calculating averages. The most well known of these is the arithmetic mean (usually just termed ‘the mean’), which is simply calculated as the sum of all values in the series divided by the number of values. The two other methods for calculating the average of a series are the mode and the median. The mode measures the most frequently occurring value in a series, which is sometimes regarded as a more representative measure of the average than the arithmetic mean. Finally, the median is the middle value in a series when the elements are arranged in an ascending order. For a symmetric distribution, the mean, mode and median will be coincident. For any non-symmetric distribution of points however, the three summary measures will in general be different. Each of these measures of average has its relative merits and demerits. The mean is the most familiar method to most researchers, but can be unduly affected by extreme values, and, in such cases, it may not be representative of most of the data. The mode is, arguably, the easiest to obtain, but it is 46 Real Estate Modelling and Forecasting not suitable for continuous, non-integer data (e.g. returns or yields) or for distributions that incorporate two or more peaks (known as bimodal and multimodal distributions, respectively). The median is often considered to be a useful representation of the ‘typical’ value of a series, but it has the drawback that its calculation is based essentially on one observation. Thus if, for example, we had a series containing ten observations and we were to double the values of the top three data points, the median would be unchanged. The geometric mean There exists another method that can be used to estimate the average of a series, known as the geometric mean. It involves calculating the Nth root of the product of N numbers. In other words, if we want to find the geometric mean of six numbers, we multiply them together and take the sixth root (i.e. raise the product to the power of 1/6th). In real estate investment, we usually deal with returns or percentage changes rather than actual values, and the method for calculating the geo- metric mean just described cannot handle negative numbers. Therefore we use a slightly different approach in such cases. To calculate the geometric mean of a set of N returns, we express them as proportions (i.e. on a (−1, 1) scale) rather than percentages (on a (−100, 100) scale), and we would use the formula R G = [ (1 + r 1 )(1 + r 2 ) (1 + r N ) ] 1/N − 1 (3.1) where r 1 ,r 2 , ,r N are the returns and R G is the calculated value of the geometric mean. Hence, what we would do would be to add one to each return, multiply the resulting expressions together, raise this product to the power 1/N and then subtract one right at the end. Which method for calculating the mean should we use, therefore? The answer is, as usual, ‘It depends.’ Geometric returns give the fixed return on the asset or portfolio that would have been required to match the actual performance, which is not the case for the arithmetic mean. Thus, if you assumed that the arithmetic mean return had been earned on the asset every year, you would not reach the correct value of the asset or portfolio at the end! It could be shown that the geometric return is always less than or equal to the arithmetic return, however, and so the geometric return is a downward-biased predictor of future performance. Hence, if the objective is to forecast future returns, the arithmetic mean is the one to use. Finally, it is worth noting that the geometric mean is evidently less intuitive and less commonly used than the arithmetic mean, but it is less affected by extreme outliers than the latter. There is an approximate relationship that holds Real estate analysis: statistical tools 47 between the arithmetic and geometric means, calculated using the same set of returns: R G ≈ R A − 1 2 σ 2 (3.2) where R G and R A are the geometric and arithmetic means, respectively, and σ 2 is the variance of the returns. 3.2.3 Measures of spread Usually, the average value of a series will be insufficient to characterise a data series adequately, since two series may have the same average but very different profiles because the observations on one of the series may be much more widely spread about the mean than the other. Hence another important feature of a series is how dispersed its values are. In finance theory, for example, the more widely spread returns are around their mean value the more risky the asset is usually considered to be, and the same principle applies in real estate. The simplest measure of spread is arguably the range, which is calculated by subtracting the smallest observation from the largest. While the range has some uses, it is fatally flawed as a measure of dispersion by its extreme sensitivity to an outlying observation. A more reliable measure of spread, although it is not widely employed by quantitative analysts, is the semi-interquartile range, also sometimes known as the quartile deviation. Calculating this measure involves first ordering the data and then splitting the sampleinto four parts (quartiles) 2 with equal num- bers of observations. The second quartile will be exactly at the halfway point, and is known as the median, as described above. The semi-interquartile range focuses on the first and third quartiles, however, which will be at the quarter and three-quarter points in the ordered series, and which can be calculated respectively by the following: Q 1 =  N + 1 4  th value (3.3) and Q 3 = 3 4 ( N + 1 ) th value (3.4) The semi-interquartile range is then given by the difference between the two: IQR = Q 3 − Q 1 (3.5) 2 Note that there are several slightly different formulae that can be used for calculating quartiles, each of which may provide slightly different answers. 48 Real Estate Modelling and Forecasting This measure of spread is usually considered superior to the range, as it is not so heavily influenced by one or two extreme outliers that, by definition, would be right at the end of an ordered series and so would affect the range. The semi-interquartile range still only incorporates two of the observations in the entire sample, however, and thus another more familiar measure of spread, the variance, is very widely used. It is interpreted as the average squared deviation of each data point about its mean value, and is calculated using the usual formula for the variance of a sample: σ 2 =  (y i − y) 2 N − 1 (3.6) Another measure of spread, the standard deviation, is calculated by taking the square root of equation (3.6): σ =   (y i − y) 2 N − 1 (3.7) The squares of the deviations from the mean are taken rather than the devi- ations themselves, in order to ensure that positive and negative deviations (for points above and below the average, respectively) do not cancel each other out. While there is little to choose between the variance and the standard deviation, the latter is sometimes preferred since it will have the same units as the variable whose spread is being measured, whereas the variance will have units of the square of the variable. Both measures share the advantage that they encapsulate information from all the available data points, unlike the range and the quartile deviation, although they can also be heavily influenced by outliers, as for the range. The quartile deviation is an appro- priate measure of spread if the median is used to define the average value of the series, while the variance or standard deviation will be appropriate if the arithmetic mean constitutes the adopted measure of central tendency. Before moving on, it is worth discussing why the denominator in the formulae for the variance and standard deviation includes N − 1 rather than N, the sample size. Subtracting one from the number of available data points is known as a degrees of freedom correction, and this is necessary as the spread is being calculated about the mean of the series, and this mean has had to be estimated as well. Thus the spread measures described above are known as the sample variance and the sample standard deviation. Had we been observing the entire population of data rather than a mere sample from it, then the formulae would not need a degree of freedom correction and we would divide by N rather than N − 1. Real estate analysis: statistical tools 49 A further measure of dispersion is the negative semi-variance, which also gives rise to the negative semi-standard deviation. These measures use identical formulae to those described above for the variance and standard deviation, but, when calculating their values, only those observations for which y i < y are used in the sum, and N now denotes the number of such observations. This measure is sometimes useful if the observations are not symmetric about their mean value (i.e. if the distribution is skewed; see the next section). 3 A final statistic that has some uses for measuring dispersion is the coefficient of variation, CV. This is obtained by dividing the standard deviation by the arithmetic mean of the series: CV = σ y (3.8) CV is useful when we want to make comparisons between series. Since the standard deviation has units of the series under investigation, it will scale with that series. Thus, if we wanted to compare the spread of monthly apartment rental values in Manhattan with those in Houston, using the standard deviation would be misleading, as the average rental value in Manhattan will be much bigger. By normalising the standard deviation, the coefficient of variation is a unit-free (dimensionless) measure of spread, and so could be used more appropriately to compare the rental values. Example 3.1 We calculate the measures of spreads described above for the annual office total return series in Frankfurt and Munich, which are presented in table 3.1. Annual total returns have ranged from −3.7 per cent to 11.3 per cent in Frankfurt and from −2.0 per cent to 13.3 per cent in Munich. Applying equation (3.3), the Q 1 observation is the fourth observation – hence 0.8 and 2.1 for Frankfurt and Munich, respectively. The third quartile value is the thirteenth observation – that is, 9.9 and 9.5. We observe that Frankfurt returns have a lower mean and higher standard deviation than those for Munich. On both the variance and standard deviation measures, Frankfurt exhibits more volatility than Munich. This is confirmed by the coefficient of variation. The higher value for Frankfurt indicates a more volatile market (the standard deviation is nearly as large as the mean return), whereas, for Munich, the standard deviation is only 0.7 times the mean return. Note that if the mean return in Frankfurt had been much higher (say 7 per cent), and all other metrics being equal, the coefficient of variation would have been lower than Munich’s. 3 Of course, we could also define the positive semi-variance, where only observations such that y i > y are included in the sum. 50 Real Estate Modelling and Forecasting Table 3.1 Summary statistics for Frankfurt and Munich returns Original data Ordered data Frankfurt Munich Frankfurt Munich 1992 4.9 2.6 −3.7 −2.0 1993 5.8 −0.1 −2.5 −0.1 1994 3.4 2.0 −0.7 2.0 1995 −0.7 −2.0 0.8 2.1 1996 −2.5 7.3 2.6 2.6 1997 5.3 7.1 3.4 4.7 1998 6.2 10.1 4.0 5.4 1999 10.4 9.5 4.9 5.6 2000 11.1 11.7 5.3 5.7 2001 11.3 5.4 5.8 7.1 2002 4.0 5.6 6.2 7.3 2003 2.6 5.7 9.6 8.0 2004 −3.7 2.1 9.9 9.5 2005 0.8 4.7 10.4 10.1 2006 9.6 8.0 11.1 11.7 2007 9.9 13.3 11.3 13.3 Min −3.7 −2.0 Max 11.3 13.3 N 16 16 Q 1 4.3 (4th) = 0.8 4.3 (4th) = 2.1 Q 3 12.8 (13th) = 9.9 12.8 (13th) = 9.5 IQR 9.1 7.4 µ 4.9 5.8 σ 2 23.0 17.8 σ 4.8 4.2 CV 0.98 0.73 Source: Authors’ own estimates, based on Property and Portfolio Research (PPR) data. 3.2.4 Higher moments If the observations for a given set of data follow a normal distribution, then the mean and variance are sufficient to describe the series entirely. In other words, it is impossible to have two different normal distributions with the same mean and variance. Most samples of data do not follow a normal Real estate analysis: statistical tools 51 distribution, however, and therefore we also need what are known as the higher moments of a series to characterise it fully. The mean and the variance are the first and second moments of a distribution, respectively, and the (standardised) third and fourth moments are known as the skewness and kur- tosis, respectively. Skewness defines the shape of the distribution, and mea- sures the extent to which it is not symmetric about its mean value. When the distribution of data is symmetric, the three methods for calculating the aver- age (mean, mode and median) of the sample will be equal. If the distribution is positively skewed (when there is a long right-hand tail and most of the data are bunched over to the left), the ordering will be mean > median > mode, whereas, if the distribution is negatively skewed (a long left-hand tail and most of the data bunched on the right), the ordering will be the opposite. A normally distributed series has zero skewness (i.e. it is symmetric). Kurtosis measures the fatness of the tails of the distribution and how peaked at the mean the series is. A normal distribution is defined to have a coefficient of kurtosis of three. It is possible to define a coefficient of excess kurtosis, equal to the coefficient of kurtosis minus three; a normal distribution will thus have a coefficient of excess kurtosis of zero. A normal distribution is said to be mesokurtic. Denoting the observations on a series by y i and their variance by σ 2 , it can be shown that the coefficients of skewness and kurtosis can be calculated respectively as 4 skew = 1 N−1  (y i − y) 3  σ 2  3/2 (3.9) and kurt = 1 N−1  (y i − y) 4  σ 2  2 (3.10) The kurtosis of the normal distribution is three, so its excess kurtosis (b 2 − 3) is zero. To give some illustrations of what a series having specific departures from normality may look like, consider figures 3.1 and 3.2. A normal distribution is symmetric about its mean, while a skewed distribution will not be, but will have one tail longer than the other. A leptokurtic distribution is one that 4 There are a number of ways to calculate skewness (and kurtosis); the one given in the formula is sometimes known as the moment coefficient of skewness, but it could also be measured using the standardised difference between the mean and the median, or by using the quartiles of the data. Unfortunately, this implies that different software will give slightly different values for the skewness and kurtosis coefficients. For example, some packages make a ‘degrees of freedom correction’, as we do in equations (3.9) and (3.10), while others do not, so that the divisor in such cases would be N rather than N − 1 in the equations. 52 Real Estate Modelling and Forecasting x f(x) x f(x) Figure 3.1 A normal versus a skewed distribution 0.5 0.4 0.3 0.2 0.1 0.0 –5.4 –3.6 –1.8 0.0 1.8 3.6 5.4 Figure 3.2 A leptokurtic versus a normal distribution has fatter tails and is more peaked at the mean than a normally distributed random variable with the same mean and variance, while a platykurtic distribution will be less peaked in the mean and will have thinner tails and more of the distribution in the shoulders than a normal. In practice, a leptokurtic distribution is more likely to characterise real estate (and economic) time series, and to characterise the residuals from a time series model. In figure 3.2, the leptokurtic distribution is shown by the bold line, with the normal by the dotted line. There is a formal test for normality, and this is described and discussed in chapter 6. We now apply equations (3.9) and (3.10) to estimate the skewness and kurtosis for the Frankfurt and Munich office returns given in table 3.1 (see table 3.2). Munich returns show no skewness and Frankfurt slightly negative skewness. Therefore returns in Munich are symmetric about their mean; in Frankfurt, however, the tail tends to be a bit longer in the negative direction. Both series have a flatter peak around their mean and thinner tails than a [...]... names for y and the xs, and all these terms are used synonymously in this book (see box 4.1) Box 4.1 Names for y and xs in regression models Names for y Names for the xs Dependent variable Regressand Effect variable Explained variable Left-hand side (LHS) variable Independent variables Regressors Causal variables Explanatory variables Right-hand side (RHS) variables 74 Real Estate Modelling and Forecasting... measure, usually known as the correlation 54 Real Estate Modelling and Forecasting coefficient, is often denoted ρx,y , and is calculated as ρx,y = σx,y (xi − x)(yi − y) = (N − 1)σx σy σx σy (3.12) where σx and σy are the standard deviations of x and y, respectively This measure is more strictly known as Pearson’s product moment correlation 3.3 Probability and characteristics of probability distributions... (a) and (b) (which show the index of US income returns for all real estate in nominal terms and the index of real office values in Tokyo, respectively) and panel (c) (which shows the allproperty returns risk premium, calculated from total returns for all real estate minus returns on medium-term UK government bonds constructed using data from the IPD) is clearly evident The series in panels (a) and (b)... standard error of the estimate √ of µ, which is the standard deviation divided by N − 1 The standard error is a measure of how confident one is in the estimate of the sample mean obtained in the first stage If a standard error is small, the value of the test statistic will be large relative to the case in which the standard error is large For a small standard error, it would not require the estimated and. .. following section 3.4 Hypothesis testing Real estate theory and experience will often suggest that certain parameters should take on particular values, or values within a given range It is therefore of interest to determine whether the relationships expected from real estate theory are upheld by the data to hand or not For example, estimates of the mean (average) and standard deviation will have been obtained... have, and the more reliable the sample average estimate will be Ceteris paribus, the larger the sample size the less evidence we would need against a null hypothesis to reject it, and so the more likely such a rejection is to occur If we take repeated samples of size N from a population that has a mean µ and a standard deviation σ , then the sample mean will be distributed with √ mean µ and standard... average transaction price of a three-bedroom 58 Figure 3.3 The normal distribution Real Estate Modelling and Forecasting f (x) x apartment in Hong Kong We could take a sample of fifty apartments that had recently been sold and calculate the mean price from them, and then another sample of the same size to calculate the mean, and so on If we did this repeatedly, we would get a distribution of mean values, with... figure 3.3, should be familiar to readers Note its characteristic ‘bell’ shape and its symmetry around the mean A normal variate can be scaled to have zero mean and unit variance by subtracting its mean and dividing by its standard deviation Real estate analysis: statistical tools 59 Table 3.3 Critical values from the standard normal versus t-distribution Significance level t40 t4 50% 5% 2.5% 0 1.64... mean and the variance of a given set of actual data It is also useful to know how to work with the theoretical expressions for the mean and variance of a random variable, however A random variable is one that can take on any value from a given set The mean of a random variable y is also known as its expected value, written E(y) The properties of expected values are used widely in econometrics, and are... variables, y1 and y2 , E(y1 y2 ) = E(y1 ) E(y2 ) The variance of a random variable y is usually written var (y) The properties of the ‘variance operator’, var, are as follows 2 ● The variance of a random variable y is given by var (y) = E[y − E(y)] ● The variance of a constant is zero: var (c) = 0 2 ● For c and d constants, var (c y + d) = c var (y) 2 ● For two independent random variables, y1 and y2 , . −3.7 2 .1 9.9 9.5 2005 0.8 4.7 10 .4 10 .1 2006 9.6 8.0 11 .1 11. 7 2007 9.9 13 .3 11 .3 13 .3 Min −3.7 −2.0 Max 11 .3 13 .3 N 16 16 Q 1 4.3 (4th) = 0.8 4.3 (4th) = 2 .1 Q 3 12 .8 (13 th) = 9.9 12 .8 (13 th). −0 .1 −2.5 −0 .1 1994 3.4 2.0 −0.7 2.0 19 95 −0.7 −2.0 0.8 2 .1 1996 −2.5 7.3 2.6 2.6 19 97 5.3 7 .1 3.4 4.7 19 98 6.2 10 .1 4.0 5.4 19 99 10 .4 9.5 4.9 5.6 2000 11 .1 11. 7 5.3 5.7 20 01 11. 3 5.4 5.8 7 .1 2002. than percentages (on a ( 10 0, 10 0) scale), and we would use the formula R G = [ (1 + r 1 ) (1 + r 2 ) (1 + r N ) ] 1/ N − 1 (3 .1) where r 1 ,r 2 , ,r N are the returns and R G is the calculated