JRF 10,2. Chenyi Hu Department of Computer Science, University of Central Arkansas, Conway, Arkansas, USA, and

The current issue and full text archive of this journal is available at wwwemeraldinsightcom/1526-5943htm JRF 10,2 142 Prediction of variability in mortgage rates: interval computing solutions Ling T He Department of Economics and Finance, University of Central Arkansas, Conway, Arkansas, USA Chenyi Hu Department of Computer Science, University of Central Arkansas, Conway, Arkansas, USA, and K Michael Casey Department of Economics and Finance, University of Central Arkansas, Conway, Arkansas, USA Abstract Purpose The purpose of this paper is to forecast variability in mortgage rates by using interval measured data and interval computing method Design/methodology/approach Variability (interval) forecasts generated by the interval computing are compared with lower- and upper-bound forecasts based on the ordinary least squares (OLS) rolling regressions Findings On average, 56 per cent of annual changes in mortgage rates may be predicted by OLS lower- and upper-bound forecasts while the interval method improves forecasting accuracy to 72 per cent Research limitations/implications This paper uses the interval computing method to forecast variability in mortgage rates Future studies may expand variability forecasting into more risk-managing areas Practical implications Results of this study may be interesting to executive officers of banks, mortgage companies, and insurance companies, builders, investors, and other financial decision makers with an interest in mortgage rates Originality/value Although it is well-known that changes in mortgage rates can significantly affect the housing market and economy, there is not much serious research that attempts to forecast variability in mortgage rates in the literature This study is the first endeavor in variability forecasting for mortgage rates Keywords Financial forecasting, Loans, Interest rates, United States of America Paper type Research paper The Journal of Risk Finance Vol 10 No 2, 2009 pp 142-154 q Emerald Group Publishing Limited 1526-5943 DOI 101108/15265940910938224 Introduction Changes in mortgage rates can significantly affect the residential real estate market, which is a considerable part of the US economy As such, mortgage rates continue to be an active research topic of interest since variability in mortgage rates represents a major risk not only to financial institutions, but also to many homebuyers as well Incorrect estimation of variability in mortgage rates can result in serious financial consequences An obvious example is the subprime mortgage problem that surfaced in 2007

Errors in pricing risks involved in mortgages cost hundreds of billions of dollars for many financial institutions across the world As a part of risk management, variability forecasting for mortgage rates can serve an important and effective tool in risk pricing for different types of financial institutions Unlike previous studies that focus on identifying influential factors on mortgage rates, the purpose of this study is to predict variability in mortgage rates Traditionally, the study of variability forecasting has been based on point data and point forecasting The point data, in fact, is a snapshot at a particular time period and consists of a unique value That is, there is only one observation at a given time period The purpose of point forecasting is to predict a future value at a specified future time period by using point data However, point forecasting is plagued with poor accuracy (Fama and French, 1997; He, 2005) To address this issue and obtain variability (interval) predictions researchers employ a variety of confidence-based interval forecasting methods Researchers use different econometric or statistical models and estimation methods, such as Bayesian, Bootstrapping, Box-Jenkins, GARCH, and Holt-Winters, to generate point forecasts, then some percent of forecasting variance is added to and subtracted from the point forecasts to get interval forecasts (Chatfield, 1993, 2001) Poor accuracy remains the major problem for those confidence-based interval forecasting approaches (Gardner, 1988; Granger, 1996) In order to enhance the accuracy for variability forecasting, researchers are beginning to explore new directions in interval forecasting He and Hu (2007) suggest that interval data may provide more information as input to generate better interval or variability forecasts Powered by interval computing, their annual stock market variability forecasting achieves an overall accuracy of more than 64 per cent (Hu and He, 2007; He and Hu, 2009a) Even with traditional rolling ordinary least squares (OLS) estimation on lower and upper bounds, He and Hu (2007) generate similar results in out-of-sample interval forecasts Encouraged by the promise of this technique for stock market variability forecasts, we apply the same interval data and interval computing method to forecast variability in another important financial area, mortgage rates The results of this paper provide additional supportive evidence for the new interval forecasting approach, and also are useful to executive officers of banks, mortgage companies, and insurance companies; builders; investors; and other financial decision-makers with an interest in mortgage rates Prediction of variability in mortgage rates 143 Literature review on factors affecting mortgage rates A recent study by Xu and Fung (2005) indicates that mortgage rates are one of several factors that impact mortgage-backed security (MBS) returns Given the fact that the MBS market is nearly $4 trillion, a method that provides better mortgage rate forecasts has enormous economic value With respect to mortgage rates, the literature indicates that macroeconomic news does have an impact on mortgage rates For example, Ramchander et al (2003) find that both inflation (proxied by the changes in the CPI) and housing starts significantly impact mortgage rates Their study also included new home sales as one of the 23 macroeconomic news events that might potentially impact mortgage rates Other studies, such as Marathe and Shawky (2003), provide evidence of the linkage between other market interest rates and mortgage rates Interestingly, their study also looks at the speed of adjustment of mortgage rates to changes in other market interest rates

JRF 10,2 144 They find that any change in market rates is reflected in mortgage rates within one month, instead of minutes or hours that it takes for stock prices to react to new information (Harvey and Huang, 1991; Ederington and Lee, 1993, 1995) Allen et al (1999) find that mortgage rates are influenced by corporate bond rates and treasury securities Their study confirms the linkage between mortgage rates and risky bonds tends to vary over time and they maintain that mortgage rates are sticky and do not adjust as rapidly as other market rates Darrat et al (1998) in their study of mortgage loan rates and deposit costs determine that mortgage rates are highly impacted by the demand for mortgages, both long- and short-term interest rates, and risk premiums In our model presented in the next section, there are several factors such as building permits, new home sales, existing home sales, and housing starts that should serve to capture the demand for mortgages He and Casey (2008), in a recent study of mortgage determinants, include these variables and find that mortgage rates are significantly impacted by new home sales and building permits While the bulk of the price variability literature focuses on market microstructure issues that investigate the speed of adjustment to new information there are studies such as He and Wang (1995) that find prices adjust in stages over multiple periods as investors often process the same new information in a differential manner since each investor holds a separate set of existing information Their study of stock prices and trading volume around information releases indicated that not all adjustment occurs immediately Fleming and Remolona (1999) provide additional support for a two-stage adjustment process that may arise from investors differential private views For this reason, we take longer-term approach and attempt to provide a better forecasting mechanism for changes in mortgage rates during the next 12 month period This type of approach is appropriate for mortgage rates since current mortgage rates are likely determined during prior months according to Ramchander et al (2003) Mortgage rates often have advance commitments, which would tend to reduce short-term volatility to new information Additionally, many investors are not in a position to take advantage of the short-term reaction of many financial assets to new information there is opportunity to take advantage of longer-term adjustments Longer time horizon forecasts will therefore be much more useful to investors of MBS, homebuilders, homebuyers, and others with interest in mortgage markets Interval forecasting is in the fledgling stages with respect to application in the financial sector However, the technique holds promise for all types of forecasts as indicated by de Gooijer and Hyndman s (2006) statement that improved forecast intervals require further research Prediction intervals by their very nature help to reduce much of the uncertainty associated with using point estimates In spite of the promise that interval methods hold for financial forecasting, there are only a few papers to date that apply this technique in this area The remainder of the paper is organized as follows In the next section, we provide a discussion of the estimation model, data, and some information on interval forecasting Then we present the model results and our conclusions appear in the last section Estimation model and data Model Most variability studies document that financial markets do respond to the release of new information with greater price volatility around the news release

In fact, Ederington and Lee (1993) document that most of the price and volatility reaction occurs within one minute of the news release Since, new information is by its very nature unpredictable, few studies have been successful in predicting future variability, particularly over longer periods of time In this study, we use interval-computing techniques to forecast mortgage rate variability The independent variables selected as mortgage rate predictors are all widely available publicly released macroeconomic variables The following seven-factor model can reveal relationships between mortgage rates, some housing factors, and other macroeconomic factors: Prediction of variability in mortgage rates 145 MORT t ¼ a t þ N t ðnhs t ÞþE t ðehs t ÞþS t ðsta t ÞþP t ðper t Þ þ C t ðcpi t ÞþF t ðdef t ÞþT t ðterm t Þþe t : ð1þ MORT represents percentage changes in mortgage rates; NHS is percentage changes in new home sales; percentage changes in existing home sales are reflected in EHS; STA and PER measure percentage changes in new housing starts and permits, respectively Both new home sales and housing starts were found to influence either bond rates, mortgage rates, or both by Ramchander et al (2003) The other housing related variables, existing home sales and building permits, help proxy mortgage demand which Darrat et al (1998) find significant in predicting mortgage demand He and Casey (2008) report that new home sales and building permits can explain a significant part of variation in mortgage rates CPI represents percentage changes in the consumer price index; and DEF and TERM are default and maturity risk premiums Ramchander et al (2003) also use the CPI and conclude that inflation does have a positive impact on mortgage rates even while the individual variable was not significant Ramchander et al (2003), Darrat et al (1998), Marathe and Shawky (2003) and He and Casey (2008) all find mortgage rates to be linked to various other market rates of interest The variables for default and maturity risks should capture those relationships There is a consensus in the literature that macroeconomic factors vary over time Many statistical methods have been introduced to deal with the time-varying issue (He, 2005) For instance, Fama and French (1997) use rolling OLS regressions to produce time-varying regression coefficients and forecasts for excess stock returns of industrial portfolios This study adopts the rolling OLS regression method to estimate time-varying coefficients Although, there are other methods to deal with the time-varying issue, a purpose of this study is to demonstrate that a simple method with interval data can produce good variability predictions In order to have enough degrees of freedom, we apply a 15-year rolling estimation period to equation (1) to get annual coefficient estimates For quarterly data, a 40-quarter rolling period is used Similar to Fama and French s (1997) out-of-sample forecasting, the rolling coefficient estimates from equation (1) are multiplied by next period s explanatory variables to get out-of-sample forecasts for the variability in mortgage rates: MORT t ¼ a t21 þ N t21 ðnhs t ÞþE t21 ðehs t ÞþS t21 ðsta t Þ þ P t21 ðper t ÞþC t21 ðcpi t ÞþF t21 ðdef t ÞþT t21 ðterm t Þ: ð2þ This study uses time series data that covers the period of January 1972-December 2005 There is no specific reason, other than the availability of data, for the selected sample period The following basic monthly series are used:

JRF 10,2 146 Mort The percentage changes in interest rates for 30-year fixed rate mortgages (Federal Home Loan Mortgage Corporation) NHS The percentage changes in new residential home sales (US Census Bureau) EHS The percentage changes in existing residential home sales (National Association of Realtors) STA The percentage changes in new housing starts (US Census Bureau) PER The percentage changes in new housing permits (US Census Bureau) LONG The monthly returns on long-term US government bonds (Stock, Bonds, Bills, and Inflation 2005 Yearbook, Ibbotson Associates) CORP The monthly returns on long-term corporate bonds (Stock, Bonds, Bills, and Inflation 2005 Yearbook, Ibbotson Associates) SHORT The monthly returns on one-month US treasury bills (Stock, Bonds, Bills, and Inflation 2005 Yearbook, Ibbotson Associates) According to Fama and French (1993), SHORT t2 1 is the proxy for the general level of expected returns on bonds CPI The percentage changes in the US consumer price index for all urbane consumers: all items The index is seasonally adjusted (Bureau of Labor Statistics) Seasonally, adjusted annual rates are used for monthly series of NHS, EHS, STA, and PER The following two series are derived from the above basic series: (1) DEF t ¼ CORP t 2 LONG t It represents the default risk premium (Fama and French, 1993) (2) TERM t ¼ LONG t 2 SHORT t2 1 It is the maturity risk premium and measures unexpected changes in interest rates (Fama and French, 1993) We use the minimum and maximum monthly numbers in a year or a quarter to construct the annual and quarterly interval data The OLS rolling method applies the lower- and upper-bound data to equations (1) and (2) to generate lower- and upper-bound forecasts, while the interval computing method directly uses the interval data and the two equations to produce interval forecasts Interval computing method We use the interval computing method and interval data to estimate equation (1) Interval computing uses the interval arithmetic (Moore, 1966) to construct an interval valued linear system of equations In interval arithmetic, the product of two intervals is defined as: ½a; bš*½c; dš ¼½min{ac; ad; bc; bd}; max{ac; ad; bc; bd}š: In order to estimate the coefficients based on the least squares principle, similar to Hu and He (2007), we need to first solve an interval linear systems of equations Ma ¼ v, where M is an 8 8 interval matrix; a is the vector of the coefficients to be determined; and v is an interval vector It is assumed that the coefficients are scalars initially By taking M mid, the mid-point matrix of M, and v mid, the midpoint vector of v, we construct a linear system of equations about a The numerical estimates of the

coefficients are obtained by using Gaussian elimination with scaled partial pivoting This initial approach has the intuition of finding an inner approximated solution (Hu et al, 2008) In order to reflect time-varying relationships better, a rolling estimation period of 15 consecutive years or 40 quarters is used to establish the interval linear system of equations The rolling coefficient estimates in intervals obtained from equation (1) are used to forecast changes of mortgage rates by estimating the out-of-sample forecasting model (equation (2)) However, the above approach has not taken the widths into considerations yet Therefore, it is essential to adjust the width of the forecasted interval Consistent with the rolling estimation period, the width is adjusted by a rolling scalar constant that is equivalent to the average width of mortgage rate intervals in the previous 15 years or 40 quarters Prediction of variability in mortgage rates 147 Accuracy of interval forecasting Variability forecasts are in the form of intervals Unlike point forecasts, which use forecasts errors to measure forecasting accuracy, interval forecasts allow a new quality measurement, accuracy ratio In the interval forecasting, there are two interesting numbers: a range covered by both forecasted and actual intervals and a maximum range stretched by the forecasted and actual intervals The first number quantifies the accurate part of an interval forecast and the second number indicates the relevant range involved The accuracy ratio is defined as the first number over the second number, that is, WðInterval > Interval est Þ=WðInterval < Interval est Þ, where W is the width function of an interval (He and Hu, 2007, 2009a; Hu and He, 2007) Of course, the accuracy ratio is zero if Interval > Interval est is empty, ie there is no overlap between the actual interval and its forecast For instance, if the predicted interval is [2, 4] and the actual interval is [1, 5], then the overlapped range is [2, 4] and the width of the overlapped interval is 2 (4-2) The maximum distance or the width of the interval [1, 5] is 4 Therefore, the accuracy ratio is 50 per cent If the predicted interval is [2, 8], the width of the overlapped interval is 3 (5-2), the maximum width is 7 (8-1), and the accuracy ratio is 3/7 Similar to He and Hu (2007), we use the OLS method and lower-bound data to predict lower bounds for changes in the mortgage rates and upper-bound data are used to forecast upper bounds Both lower- and upper-bound data are simultaneously used in the interval computing method to generate interval forecasts Accuracy ratios for the two methods can reveal the quality difference between the OLS lower- and upper-bound forecasts and the interval forecasts Empirical results Table I provides summary statistics for all variables All lower-bound means are negative and upper-bound means are positive, except for CPI The quarterly data have larger coefficients of variation for all variables, compared to the annual data For example, quarterly coefficients of variation for mortgage rates are more than doubled than annual ones for both lower and upper bounds The higher variability can adversely affect forecasting quality Results in Table II suggest that on average, 56 per cent of annual changes in mortgage rates may be predicted by OLS lower- and upper-bound forecasts over the period of 1987-2005

JRF 10,2 148 Table I Summary statistics (in per cent) for interval data MORT NHS EHS START PERMIT CPI DEF TERM Panel A: annual intervals (1972-2005) Lower-bound mean 2376 21063 2579 21240 2918 007 2167 2453 SD 223 445 300 547 539 029 103 208 Coef of variation 2059 2042 2052 2044 2059 414 2062 2046 Upper-bound mean 483 1214 670 1443 973 073 165 501 SD 339 587 266 671 624 037 099 293 Coef of variation 070 048 040 047 064 051 060 058 Panel B: quarterly intervals (19721-20054) Lower-bound mean 2174 2575 2289 2645 2432 025 2084 2202 SD 212 479 310 526 525 030 092 247 Coef of variation 2122 2083 2107 2082 2122 120 2110 2122 Upper-bound mean 183 664 349 736 472 054 085 250 SD 299 574 309 632 512 032 092 258 Coef Of variation 163 086 089 086 108 059 108 103 Notes: MORT percentage changes in mortgage rates; NHS percentage changes in new home sales; EHS percentage changes in existing home sales; CPI percentage changes in consumer price index; START percentage changes in new home starts; PERMIT percentage changes in housing permits; DEF default risk premium; TERM unexpected changes in interest rates The standard deviation of accuracy ratios for 15 annual forecasts is 216 per cent A year-to-year comparison of the predicted and actual variability in mortgage rates is shown in Figure 1 Overall, the actual variability of mortgage rates is smaller than the forecasted variability Another interesting observation is that the OLS upper-bound forecasts are more unstable than lower-bound forecasts Consistent with findings reported by He and Hu (2007), these results may disclose a major weakness of the OLS lower- and upper-bound forecasting method, the separation of lower- and upper-bound data That is, the process of generating lower-bound forecasts is completely separated from upper-bound forecasting It is obvious that the variability information contained in the interval data is not fully used in lower- and upper-bound forecasting On the other hand, an important strength of the interval computing method is to use lower- and upper-bound data simultaneously in its computing process, therefore, variability information is fully utilized As a result, variability or interval forecasts produced by the interval computing are significantly more accurate and consistent than the OLS lower- and upper-bound forecasts The mean of accuracy ratios for interval forecasts is as high as 72 per cent and the standard deviation of accuracy ratios is only 134 per cent (Table II) The traditional forecasting quality measure, forecast error, tells the same story Both mean and standard deviation of forecast errors for interval forecasts are less than half of that for the OLS lower- and upper-bound forecasts The differences are statistically significant It is worthy to note that in contrast to the OLS lower- and upper-bound forecasts, variability forecasted by the interval computing is smaller than the actual variability in mortgage rates (Figure 2) When the more volatile quarterly interval data are used, the less accurate forecasts are generated by both methods The average of accuracy ratios reduces to 45 per cent for the interval computing and 373 per cent for the OLS method Nevertheless, the difference in accuracy for the two methods is significant The poorer accuracy of the

Interval computing Test result Panel A: annual forecasts (1987-2005) OLS interval Number of zero accuracy 0 0 Forecasting accuracy (FA) 072177 055997 SD of FA 013444 021554 t-statistic 278 * F-statistic 257 *** Forecast error mean (IE) 003052 006390 SD of IE 002026 005081 t-statistic 2266 ** F-statistic 629 * Panel B: quarterly forecasts (19821-20054) OLS interval a Number of zero accuracy 0 4 Forecasting accuracy (FA) 044939 037310 SD of FA 020597 021510 t-statistic 251 ** F-statistic 109 Forecast error mean (IE) 002996 003748 SD of IE 001561 002051 t-statistic 2286 * F-statistic 173 *** Notes: *, **, *** are significant at: 1, 5, and 10 per cent levels, respectively a There are five lower-bound forecasts greater than upper-bound forecasts in the following quarters: 19824, 19831, 19832, 20014, and 20033 We reversed those misspecified bounds to calculate accuracy ratios and forecast errors Zero accuracy a forecasted interval that does touch the actual interval; forecast error absolute value of the difference between the forecasted and actual values FA the range covered by both the forecasted and actual intervals divided by the maximum range stretched by the two intervals The t-statistic tests the null hypothesis of equality of means without the assumption of equal population variance The F-statistic tests the null hypothesis of equality of variances Prediction of variability in mortgage rates 149 Table II Out-of-sample forecasts based on interval computing and OLS lower- and upper-bound estimates OLS forecasts is partially the result of four zero accuracy ratios that indicating no overlapped areas at all by predicted and actual intervals (Table II) These zero accuracy ratios occur in the following quarters: 19842, 19863, 20001, and 20044 Another potential problem in the OLS lower- and upper-bound forecasting is that a forecasted lower bound may be larger than a predicted upper bound Five OLS quarterly forecasts have this misspecification problem (see the last note in Table II) The separation of lower and upper bounds in the OLS estimating process may be the main reason for causing misspecification in lower- and upper-bound forecasts The mean and standard deviation of forecast errors for the OLS lower and upper forecasts are also significantly higher than those for interval forecasts Compared with the annual results, the standard deviation of quarterly accuracy ratios (2060 per cent) is much higher than the annual one (1344 per cent); while for the OLS lower- and upper-bound forecasts, the annual and quarterly standard deviations of accuracy

JRF 10,2 025 02 OLS lower OLS upper Mort lower Mort upper 015 150 Changes in mortgage rates 01 005 0 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 005 Figure 1 OLS lower and upper 01 bound forecasts vs actual intervals of mortgage rates 015 Time 014 012 Lower forecast Upper forecast Mort lower Mort upper 01 008 Changes in mortgage rates 006 004 002 002 Figure 2 004 Interval forecasts vs actual intervals of 006 mortgage rates 008 0 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Time ratios are almost identical The results beg a more rigorous test on the relationship between volatility and forecasting quality for the two methods We divide 19 annual forecasts into two groups: nine years with the less volatile mortgage rates and ten more volatile years For the less volatile years, the average accuracy ratio is 7142 per cent for the interval computing and 5117 per cent for the OLS lower- and upper-bound forecasts; the forecast errors are 258 per cent vs 697 per cent The differences are statistically significant (Table III) For the ten more volatile years, there is a marginal increase in the average accuracy ratio (7286 per cent) for the interval forecasts; however, the forecast error increases to

Interval computing OLS interval Test result Panel A: nine less volatile years of the sample of 19 years Number of zero accuracy 0 0 Forecasting accuracy (FA) 071415 051166 SD of FA 012493 012958 t-statistic 337 * F-statistic 106 Forecast error mean (IE) 002584 006970 SD of IE 001089 005359 t-statistic 2241 ** F-statistic 2422 * Panel B: ten more volatile years of the sample of 19 years Number of zero accuracy 0 0 Forecasting accuracy (FA) 072862 060345 SD of FA 014888 027122 t-statistic 128 F-statistic 332 *** Forecast error mean (IE) 003474 005868 SD of IE 002595 005046 t-statistic 2133 F-statistic 378 *** Notes: *, **, *** are significant at: 1, 5, and 10 per cent levels, respectively Zero accuracy a forecasted interval that does touch the actual interval; forecast error absolute value of the difference between the forecasted and actual values FA the range covered by both the forecasted and actual intervals divided by the maximum range stretched by the two intervals The t-statistic tests the null hypothesis of equality of means without the assumption of equal population variance The F-statistic tests the null hypothesis of equality of variances Prediction of variability in mortgage rates 151 Table III Annual forecasting quality in different periods 347 per cent On the other hand, there is a decrease in the forecast error for the OLS lower- and upper-bound forecasts (587 per cent), while the average accuracy ratio increases to 6035 per cent As a result, between the two methods, there are no significant differences in the accuracy ratios and forecast errors (Table III) The results suggest that interval forecasts are consistent and stable in either volatile or stable years Unlike interval forecasts, the OLS lower- and upper-bound forecasts can enhance forecasting quality in volatile years at a cost of higher forecasting instability The interval forecasts for the ten more volatile years have significantly lower standard deviations of accuracy ratios and forecast errors than the OLS lower- and upper-bound forecasts Table IV reports quarterly forecasts in 48 more or less volatile quarters The results are consistent with annual forecasts In the less volatile quarters, interval forecasts enjoy significantly higher accuracy and lower forecast errors than the OLS lower- and upper-bound forecasts However, there is a minor increase in forecast errors and a decrease in forecast accuracy for interval forecasts in the more volatile quarters On the other hand, the OLS method raises accuracy to 4107 per cent for the lower- and upper-bound forecasts As a result, the higher accuracy ratio (4356 per cent) and lower

JRF 10,2 152 Table IV Quarterly forecasting quality in different periods Interval computing OLS interval Test result Panel A: 48 less volatile quarters of the sample of 96 quarters Number of zero accuracy 0 3 Forecasting accuracy (FA) 046322 033551 SD of FA 012372 019963 t-statistic 303 * F-statistic 115 Forecast error mean (IE) 002671 003799 SD of IE 001194 001667 t-statistic 2381 * F-statistic 195 ** Panel B: 48 more volatile quarters of the sample of 96 quarters Number of zero accuracy 0 1 Forecasting accuracy (FA) 043556 041068 SD of FA 019920 022535 t-statistic 057 F-statistic 128 Forecast error mean (IE) 003321 003698 Std dev of IE 001811 002391 t-statistic 2087 F-statistic 174 *** Notes: *, **, *** are significant at: 1, 5, and 10 per cent levels, respectively Zero accuracy a forecasted interval that does touch the actual interval; forecast error absolute value of the difference between the forecasted and actual values FA the range covered by both the forecasted and actual intervals divided by the maximum range stretched by the two intervals The t-statistic tests the null hypothesis of equality of means without the assumption of equal population variance The F-statistic tests the null hypothesis of equality of variances forecast error mean (332 per cent) for the interval forecasts are not significantly different from those for the OLS lower- and upper-bound forecasts The only significant difference is in the standard deviation of forecast errors, 181 per cent for the interval forecasts vs 239 per cent for the OLS lower- and upper-bound forecasts Concluding comments Changes in mortgage rates can affect millions of American households and different financial institutions We developed a seven-factor model to quantify relationship between mortgage rates and seven macroeconomic variables All variables are measured in intervals The interval computing method and a more traditional method, the OLS, are used to analyze the interval data and generate out-of-sample interval (variability) forecasts for mortgage rates Overall, results suggest that variability of mortgage rates may be predictable On the average, this study finds that 72 per cent of annual variability in mortgage rates can be predicted by interval forecasts over a period of 1987-2005 The OLS lower- and upper-bound forecasts can predict 56 per cent of variability in mortgage rates When more unstable quarterly data are used,

the average accuracy drops to 45 per cent for the interval forecasts and 37 per cent for the OLS lower- and upper-bound forecasts Results of this study indicate that forecasting quality is higher for the interval computing than the OLS lower- and upper-bound forecasting The interval forecasts not only have significantly higher accuracy but also significantly lower average forecast error than the OLS lower- and upper-bound forecasts In general, the interval forecasts are much stable than the OLS lower- and upper-bound forecasts, in terms of smaller standard deviations of accuracy ratios and forecast errors The difference in data processing may be the ultimate reason for the quality difference between the two methods The interval-computing method uses whole interval information to generate interval forecasts, while the OLS method uses lower- and upper-bound information separately to produce lower- and upper-bound forecasts This separation of lower and upper bounds means loss of variability information Our results also suggest that the interval computing approach can make consistent and stable interval forecasts in different volatile periods In contrast, the OLS method can produce more accurate lower- and upper-bound forecasts at a cost of higher instability Variability is one of the most critical aspects of financial system In order to properly price and manage risk for different financial products, it is important to improve the quality of variability forecasting Therefore, the current research can be easily extended to different types of financial institutions and markets Furthermore, developing alternative estimation techniques, such as the midpoint method of He and Hu (2009b), is definitely a promising research direction in variability forecasting Prediction of variability in mortgage rates 153 References Allen, M, Rutherford, R and Wiley, K (1999), The relationships between mortgage rates and capital-market rates under alternative market conditions, Journal of Real Estate Finance & Economics, Vol 19, pp 211-21 Chatfield, C (1993), Calculating interval forecasts, Journal of Business and Economic Statistics, Vol 11, pp 121-44 Chatfield, C (2001), Prediction intervals for time-series forecasting, in Armstrong, JS (Ed), Principles of Forecasting: Handbook for Researchers and Practitioners, Kluwer Academic, Boston, MA, pp 475-94 Darrat, A, Dickens, R and Glascock, J (1998), Mortgage loan rates and deposit costs: are they reliably linked?, Journal of Real Estate Finance & Economics, Vol 16, pp 27-42 de Gooijer, J and Hyndman, R (2006), 25 years of time series forecasting, International Journal of Forecasting, Vol 22, pp 443-73 Ederington, L and Lee, J (1993), How markets process information: news releases and volatility, The Journal of Finance, Vol 48, pp 1161-91 Ederington, L and Lee, J (1995), The short-run dynamics of price adjustment to new information, Journal of Financial and Quantitative Analysis, Vol 30, pp 117-34 Fama, E and French, K (1993), Common risk factors in the returns of stocks and bonds, Journal of Financial Economics, Vol 33, pp 3-56 Fama, E and French, K (1997), Industry costs of equity, Journal of Financial Economics, Vol 43, pp 153-93

JRF 10,2 154 Fleming, M and Remolona, E (1999), Price formation and liquidity in the US treasury market: the response to public information, The Journal of Finance, Vol 54, pp 1901-15 Gardner, E (1988), A simple method of computing prediction intervals for time series forecasts, Management Science, Vol 34, pp 541-6 Granger, CWJ (1996), Can we improve the perceived quality of economic forecasts?, Journal of Applied Econometrics, Vol 11, pp 455-73 Harvey, C and Huang, R (1991), Volatility in the foreign currency futures market, The Review of Financial Studies, Vol 4, pp 543-69 He, H and Wang, J (1995), Differential information and dynamic behavior of stock trading volume, The Review of Financial Studies, Vol 8, pp 919-72 He, LT (2005), The instability and predictability of factor betas and industrial stocks: the flexible least squares solutions, The Quarterly Review of Economics and Finance, Vol 45, pp 619-40 He, LT and Casey, KM (2008), A note on the risk sensitivities of mortgage rates: the impact of FIRREA, working paper He, LT and Hu, C (2007), Impacts of interval measurement on studies of economic variability: evidence from stock market variability forecasting, The Journal of Risk Finance, Vol 8, pp 489-507 He, LT and Hu, C (2009a), Impacts of interval computing on stock market forecasting, Computational Economics, Vol 33 (in press) He, LT and Hu, C (2009b), Midpoint method and accuracy of variability forecasting, Empirical Economics, 26 November 2008(in press) Hu, C and He, LT (2007), An application of interval methods for stock market forecasting, Journal of Reliable Computing, Vol 13, pp 423-34 Hu, C, Kearfott, B, Korvin, A and Kreinovich, V (2008), Knowledge Processing with Interval and Soft Computing, Springer, London Marathe, A and Shawky, H (2003), The structural relation between mortgage and market interest rates, Journal of Business Finance & Accounting, Vol 30, pp 1235-52 Moore, RE (1966), Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ Ramchander, S, Simpson, M and Webb, J (2003), Macroeconomic news and mortgage rates, Journal of Real Estate Finance & Economics, Vol 27, pp 355-77 Xu, X and Fung, H (2005), What moves the mortgage-backed securities market?, Real Estate Economics, Vol 33, pp 397-426 Corresponding author Ling T He can be contacted at: linghe@ucaedu To purchase reprints of this article please e-mail: reprints@emeraldinsightcom Or visit our web site for further details: wwwemeraldinsightcom/reprints

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