forecasting with bubblesweb.hec.ca/scse/articles/tkacz.pdf · 2005-05-10 · forecasting with...

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Forecasting with Bubbles

Greg Tkacz and Carolyn Wilkins‡ May 6, 2005

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PRELIMINARY AND INCOMPLETE DRAFT, DO NOT QUOTE WITHOUT

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Abstract This study looks at equity and housing prices and simple measures of price misalignments for Canada to determine whether there is leading information for output growth and inflation. Stock and Watson (2003) conclude that the information content of asset prices in general, and equity and housing prices in particular, are unreliable in that they do not systematically predict future economic activity or inflation. The analysis in Stock and Watson is restricted to simple linear relationships that do not pick up the potential nonlinear effects of asset price misalignments. This paper tests for nonlinearities using a two-regime threshold model. Preliminary findings are that stock and housing prices can be of use in predicting output growth particularly at medium -run forecast horizons. Our findings also show that non-linearities in the relationships between asset prices and output growth are most prevalent at the 4 and 8-quarter forecasting horizons, indicating that users of asset prices as indicators of output growth could exploit such nonlinearities to improve their forecasts. These results will be subjected to a series of robustness checks. JEL Classification Numbers: C53, E44

‡ Bank of Canada, Department of Monetary and Financial Analysis. Email: [email protected] , [email protected] The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada. This paper shall not be reproduced without the conventional permission of the Bank of Canada.

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Table of Contents

1. Introduction 2. Related Literature 3. Empirical Methods 4. Data

5. Results 6. Conclusions

References

Annexes:

Annex 1: BEAM measure of stock price misalignments

Annex 2: In-Sample Linear Regression Results

Annex 3: Estimated Thresholds

Annex 4: Linear Models: Out-of-Sample Forecast Performance

Annex 5: Threshold Models : Out-of-Sample Forecast Performance

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1. Introduction Asset prices play an important role in the monetary transmission mechanism, because, among other things, they determine the value of wealth. Asset prices also determine the value of collateral posted by households and firms to obtain loans from banks. Finally, one important asset price, namely house prices, enters the calculation of the price of housing services in the CPI and so affect inflation directly. These prices are also interesting to policymakers because of their potential to signal future developments in key target variables, inflation and output. It is clear from Charts 1 and 2 that real stock prices and real housing prices in Canada, as in many countries, are highly correlated with the business cycle. The issue of how to respond to movements in asset prices has gained prominence over the last decade, following an increasing number of asset price booms and busts in many countries that have successfully achieved a low and stable inflation environment (Borio and White, 2003; Goodhart, 2003). The debate of how a central bank should best react to asset prices is focused in large part on two issues: the ability of the monetary authority to identify asset price misalignments, and the usefulness of asset prices and measures of price misalignments in signaling future economic developments of interest to the policymaker. 1 Chart 1: Chart 2:

This paper seeks to inform the latter aspect of this debate by assessing whether asset prices and simple measures of price misalignments can predict GDP growth and inflation in Canada over horizons relevant for policy. We also test whether the relationships are characterized by non-linearities, as one would expect in an economy with financial frictions, using a two-regime threshold model. There are many important asset prices in the economy, but this paper will focus on equity and housing prices. These prices are worth special attention given their large

1 See Borio and White (2004) and Bean (2003) for excellent reviews of the issues. See Selody and Wilkins (2004) for a review of the issues from a Canadian perspectiv e.

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share in household and business balance sheets and the fact that these asset prices have historically been prone to episodes of volatility and misalignments (i.e., bubbles2). The National Balance Sheet data indicate that household assets in Canada are heavily weighted towards real assets such as housing, accounting for around 46 per cent of household wealth in 2003 (Chart 3). Financial assets such as life insurance, pensions, equities and mutual funds make up the balance.3 Once liabilities are accounted for and adjustments are made to estimate the market value of assets Macklem (1994) 4, however, the stock market accounts for around half of net non-human wealth (Chart 4). Exchange rates are also subject to this kind of volatility but are not considered in this paper. Chart 3: Chart 4:

Interpreting the evidence as to whether asset prices are useful indicators for monetary policy leaves ample room for debate. On the one hand, Borio and Lowe (2004) find that simple measures of misalignment in credit markets are useful indicators of economic downturns and banking stress. On the other hand, Stock and Watson (2003) find that among the G-7 countries asset prices in general, and equity and housing prices in particular, are unreliable indicators for monetary policy. These authors do not rule out the possibility that a stable non-linear relationship between asset prices and economic activity exists, although they do not test for this. Some promising evidence exists for the U.S. showing non-linear relationships between economic growth and measures of misalignments in equity markets (Chauvet, 1999; Bradley and Dennis, 2004). The remainder of this pape r is organized as follows. Section 2 further articulates the research question in the context of the relevant literature. Sections 3 and 4 present the empirical methods and data. The empirical results from the linear and non-linear

2 The terms misalignment and bubble are used interchangeably in reference to any deviation of an asset price from its fundamental value. 3 These data, although used extensively in analysis and international comparisons, do not properly account for changes in the market value of these assets over time. As a result, the share of a particular asset is likely (under) over -stated when its relative price is (rising) falling. 4 The market value of equity is estimated by augmenting the book value by the increase in the TSX. A similar procedure is applied to housing assets using the MLS price index.

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analysis are presented in Section 5. Section 6 concludes suggests avenues for future research. 2. Related Literature If fluctuations in asset prices contained leading information about GDP and/or inflation then they should be included in the information set considered by policymakers. In theory, if asset markets are informationally-efficient and market participants are rational, then asset prices should reflect relevant information existing elsewhere about expected future events (Smith 1999). This would be true, for instance, if equity prices were the discounted present value of future dividends and housing prices were the present discounted value of imputed rents. In practice, however, financial markets are incomplete, information is costly to acquire and not all movements in asset prices reflect changes in fundamentals such as rational estimates of discount rates and future dividend streams. Therefore, asset price movements may send signals about future economic developments that are not contained in standard indicators used by policymakers. This may especially true when asset prices are driven by non-fundamental factors or irrational behaviours. 5 For many countries the empirical evidence suggests that the information content of asset prices in general, and equity and housing prices in particular, is unreliable or underwhelming. 6Stock and Watson (2003) assess with linear methods the relative information content of 38 indicators from seven developed economies including Canada, finding that the predictive power of asset prices for output growth and inflation is varies between countries and tends to be unstable over time. Other researchers have found that some useful information for monetary policy can be extracted from housing and equity prices using linear methods in some countries. For example, Goodhart and Hofmann (2000) find that housing prices have leading indicator properties for inflation in 12 countries; although Cecchetti et al. (2000) and Filardo (2000) show that the inclusion of housing prices does not improve inflation forecasts in an economically significant manner. Work on a financial conditions index (FCI) for Canada, which includes housing and equity prices, provides some leading information for output at some horizons but not for inflation (Gauthier, Graham and Li (2004)). While this work suggests a promising avenue for future research, overall the evidence suggests that indicator models that include these types of asset prices should not receive a large weight in policy decisions. A possible explanation for the disappointing performance of asset prices as indicators may be that linear, reduced-form techniques are inadequate for capturing the underlying relationship between asset prices and the real economy. Gilchrist and Leahy 2002), among others, suggest that movements in asset prices should be evaluated 5 This paper does not address the question of why asset prices may deviate from fundamentals. See Selody and Wilkins (2004) for a brief general discussion of the literature on bubbles and Sushil Bikhchandani and Sunil Sharma (2000) for an excellent review of financial market herding models. 6 Performance measures are typically based on comparisons of out-of-sample forecasts at different horizons relative to a simple au toregressive model.

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only in structural-behavioral models that are explicit about their causal or structural relationship to economic activity. 7 Alternatively, Stock and Watson (2003) suggest the possibility that a stable non-linear relationship exists between asset prices and economic activity, although they do not test this hypothesis. 8 One reason why asset prices may exhibit a non-linear relationship with the real activity is that financial frictions cause agents to face financing constraints. As the net wealth increases with a rise in asset prices these financing constraints are relaxed so that more projects can be financed. This effect diminishes to zero, however, as asset prices rise by enough to ensure that all agents are no longer constrained (Kiyotaki and Moore, 1997). A related explanation is that when prospects for future productivity growth rise, the value of the firm increases and financing constraints are relaxed, even if current growth rate is unchanged (Jermann and Quadrini, 2003). Alternatively, the relationship between asset prices and economic activity may be nonlinear if asset price misalignments, or bubbles, have different empirical properties than asset price fluctuations driven by fundamentals. This may be because the underlying behaviours that drive asset price bubbles than those that drive asset price fundamentals (Filardo, 2001). For example, bubbles may reflect periods where expectations become extrapolative while non-bubble periods are characterized by mean-reverting expectations (Shiller, 2003). There exists some empirical evidence of non-linear relationships between economic growth and measures of misalignments in asset prices, although this literature is in early days and tends to focus more on equity markets than on housing markets. For example, Bradley and Dennis (2004) study whether unusual changes in stock returns (and excess returns) have any information for U.S. output over the 1934 to 2002 period. The authors reject linearity and find interesting threshold effects, although out of sample forecasting is poor relative to the linear model due to over fitting. Chauvet (1999) tests numerous stock market factors (e.g., excess stock returns, SP500 dividend yield) as predictors of business cycle turning points, and finds that stock market factors perform better than typical business cycle indicators, even in real time. Borio and Lowe (2003) find a significant relationship between several measures of financial imbalances and banking distress, as well output and inflation declines up to 4 years ahead. Tkacz (2001) finds improvements in forecasting Canadian GDP using neural network models, while Tkacz (2002) finds evidence of threshold effects in the relationship between the yield spread and inflation in the U.S. and Canada. Galbraith and Tkacz (2000), however, find only limited international evidence of a non-linear relationship between the term spread and output among G-7 countries.

7 The authors refer to arguments made by Woodford (1994) that poor forecasting performance of an indicator may be expected i f policy makers use this information and respond to it. 8 An alternative reason for the instability may be the changing nature of financial structures within countries across time, or the differing types of financial structures across countries, although this latter explanation does not seem to be validated by investigation of the data for 29 OECD countries (Djoudad, Selody and Wilkins (2004)).

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This paper seeks to add to this literature by looking first at the predictive power for GDP and inflation of equity and housing prices, and simple measures of misalignments of these prices, and then test for non-linearity in the form of threshold effects in these relationships. (Trying to explain the value-added here) 3. Empirical Methods The methodology is designed to answer the following questions: Do measures of asset prices and asset price misalignments have useful information for monetary policy? Is this information better extracted using a linear model or a model with threshold effects? In order to answer these questions we first test using linear techniques whether asset prices and simple measures of asset-price misalignments can help predict output and inflation over forecast horizons spanning one to 16 quarters. This assessment is based on both in and out of sample criteria. We then test for non-linearities in the relationships using a two-regime threshold model, and test the out-of-sample forecasting performance of these relationships. 3.1 Linear Models

The linear approach assumes that the target variables are linear functions of the indicator variables, according to the following general equation of the Stock and Watson (2003) type:

tktkktktk Xyy ξγβα +++= −− ,,, . (1)

tky , is the target variable of interest in cumulative growth rate terms (annual rates)

over k period where k = 1, 4, 8, 12 and 16 quarters. Lagged values of ky are included as explanatory variables to account for serial correlation and to avoid misspecification problems. ktkX −, is a vector of indicator variables of interest. Two benchmark models are defined for each of the target variable. Benchmark model 1 (BM1) is a simple AR1 model. Benchmark model 2 (BM2) for CPI inflation is the AR1 augmented by the output gap (a pseudo Phillips Curve), while BM2 for real output growth is the AR1 augmented by the output gap and the term spread (a pseudo IS curve). BM1 allows comparisons with results in the literature, while BM2 tests the value-added of the asset price variables relative to other variable typically used by policy-makers. Equation 1 is estimated first by adding each asset price measure to each of the benchmarks, and then in combinations. The Newey and West (1987) correction is applied to the variance-covariance matrix of the residuals to correct the error term for serial correlation and heteroscedasticity by calculating a consistent variance-covariance matrix. These augmented equations are then compared with each of the benchmarks

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using both in-sample and out-of-sample criteria. The out-of-sample forecasts are constructed by estimating the model to period t (1999:Q4) and then produce forecasts for period t+k; the sample is then updated so that model is re -estimated using data up to t+1, and a forecast is then produced for t+k+1. We continue in this fashion until out-of-sample forecasts are generated for the full out-of-sample period 2000:Q1 to 2004:Q4. 3.2 Threshold Models We next estimate a model that allows for asymmetry in the form of a threshold effect of asset prices on output growth and inflation. As such, the relationship between asset prices and macro variables will be dependent either on the magnitude of the deviation of an asset price from its fundamental value, or on the magnitude of the growth rate of the asset price. The models take the form

tktkktkktktk zXyy ξδγβα ++++= −−− ,1

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, for τ≤− ktkz , (2)

tktkktkktktk zXyy ξδγβα ++++= −−− ,2

,2

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, for τ>− ktkz , (3) where z is some variable extracted from the vector X, usually representing stock or housing prices, and t represents the level of z that triggers a regime change. Superscripts denote the values taken in regimes 1 and 2, respectively. The threshold level is unknown a priori, so we perform a grid search over 200 different values of the threshold variable in an attempt to maximize the probability of locating a significant threshold, should it indeed exist. Consistent with Andrews (1993), we trim the grid by 15% at each end in order to minimize the effects of outliers. To conduct inference on the significance of the threshold, we use the bootstrap procedure proposed by Hansen (1996). The estimated thresholds for each model, along with their accompanying p-values are computed using 2000 bootstrap replications. Similar strategies were employed by Galbraith and Tkacz (2000) and Tkacz (2004) for mapping the relationship between the term spread and either GDP growth or inflation. Note that for some models (13 through 16) we have two candidate threshold variables, and so test the significance of each in turn. This therefore yields 20 different models for each horizon k.

To estimate the parameters of the threshold model (2)-(3), we follow Hansen (2000) who derives an approximation of the asymptotic distribution of the least squares estimator of the threshold parameter τ̂ . To understand how the parameters are estimated, we introduce an indicator function d and can re -write equations (2) and (3) as a single equation:

tktkktkktkktkktkktktk DdzCdXBdyAdzXyy ξδγβα ++++++++= −−−−−− ,,,,2

,2

,22

, (4) where

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>≤

=−

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ττ

ktk

ktk

zz

d,

,

01

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12 αα =+ A , 12 ββ =+ B , 12 γγ =+ C , and 12 δδ =+ D . By assuming that τ̂ is bounded by the largest and smallest values of the asset price variables, we can estimate the parameters in (4) by least squares conditional on a given value of τ̂ . By iterating through the possible values of τ̂ in the range of available asset price growth rates or deviations from equilibrium, we select the τ̂ that minimizes the sum of squared residuals in (4) and are therefore not constrained by the trimming used to conduct inference on the existence of a threshold. To perform out-of-sample forecasts using the threshold model, we estimate the model, both the parameters and the threshold, up to period t (1999:Q4) and then produce forecasts for period t+k; the sample is then updated so that model is re-estimated using data up to t+1, and a forecast is then produced for t+k+1. We continue in this fashion until out -of-sample forecasts are generated for the full period 2000:Q1 to 2004:Q4. 4. Data This study uses quarterly Canadian data over the period 1981:Q1 to 2004:Q4 for real GDP, consumer price inflation (all items), the TSX index, and t he new housing price index. Equity and housing prices are deflated by the consumer price index (CPI). These variables are transformed into cumulative, annualized growth rates over the k horizons. The term spread is defined as the difference between the 10 -year-and-over government bond yield less the 90-day commercial paper rate. The output gap is taken from the Quarterly Projection Model (QPM) (Coletti et al. 1996) 9 The measures of misalignment in asset prices are proxied by deviation from trend as measured by a Baxter-King filter of the levels of the TSX and the new housing price index. 10 An additional measure of misalignment in equity prices is taken from Gauthier and Li (2004). 11 The details of the Gauthier and Li measure are in Annex 1. Any definition of an asset price misalignment is highly subjective since there are many different yet legitimate ways to think about fundamental value. Taking a mechanical view of fundamentals is popular in the empirical literature (e.g., Helbling and Terrones, 2003; Detken and Smets 2003), mainly because it is relatively straightforward to apply to large data sets and does not rely on any particular theory. Details of the mechanics differ between studies but the premise is the same: asset price fundamentals are captured by a slow-moving trend line through the time line mapped out by actual asset prices. For example, Detken and Smets (2003) use a one-side Hodrick-Prescott filter to

9 The same equations were estimated using CPIX with generally poorer results than with CPI. The existing price index was also tested, but not reported here because of short sample size. 10 This measure has the disadvantage of including future information, and therefore is considered to be a rough first -pass. Given the encouraging results, the authors are currently working on replacing this with more satisfactory filter-based measures of misalignment. 11 All series used in the empirical analysis appear to be stationary by at least one standard unit root test.

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determine fundamental values of the asset prices in their study, which includes housing and stock prices. The disadvantage of a mechanical approach, like the one employed in this study, is that there is no explicit link between the trend line (i.e., the fundamental value) and the underlying economic forces that may be responsible for the movements in asset prices, and so the estimate of fundamental value is ad hoc. Defining an asset price bubble without any reference to a structural model that delineates the underlying behaviours that cause the bubble is somewhat problematic. The problem arises because any evidence of a bubble can be attributed to a bad estimate of fundamental value: attributable to variables that are unobserved by researchers and therefore missing from the model of fundamental value (Hamilton and Whiteman, 1985). An alternative mechanical definition of asset price bubbles is given by Kindleberger (1987), and can be found in the New Palgrave Dictionary of Money and Finance. It is based on the rate of change of an asset price, rather than its deviation from fundamental value: a bubble is “a sharp rise in the price of an asset or a range of assets in a continuous process, with the initial rise generating expectations of further rises and attracting new buyers—generally speculators—interested in profits from trading in the asset rather than its use or earning capacity.” Case and Shiller (2003) find evidence in the U.S. housing market of this type of extrapolative expectations. 12 Under this definition, the real growth rates in asset prices could also be considered to capture a bubble component.13 This approach has the advantage of relying only on observed data, although it may be capturing effects other than those coming from bubbles. A method that does link fundamental value to macroeconomic variables is to use a macroeconomic model to identify the long-run determinants of an asset price such as equity or housing prices. The fundamental value in this approach is defined as the accumulation of permanent shocks to asset prices (Dupuis and Tessier, 2003; Gauthier and Li, 2004). 14 We use the measure of misalignment for the TSX from the Bonds Equity and Money (BEAM) model in Gauthier and Li (2004). Unfortunately no similar measure exists for house prices in Canada. 15 While this approach to determining fundamental value has the appeal of being linked to the macro econo my, the link is reduced-form without macroeconomic behavioural foundations. It also uses econometric estimates rather than arbitrary exogenous assumptions about the future path of revenue streams and discount rates to determine fundamental values as is the case in standard valuation models. The weakness of this approach is that there is no

12 In a survey of the expected housing price increase over the next decade by U.S. house buyers , the authors observe that even after the recent long boom in U.S. house prices buyers are still expecting double-digit average annual price increases over the next decade. 13 For example, Bordo and Jeanne (2002) use this type of definition, basing their measures of asset prices booms and busts on abnormal growth rates rather than on the percentage deviation from a slow-moving trend line that represents fundamentals. 14 Alternatively, the fundamental value could be defined as the price of the asset predicted by the cointegrating vector, so that t he asset price misalignment is the deviation of the actual asset price from the price predicted by the cointegrating vector. 15 Traclet (2005) estimates measures of misalignment in regional housing prices in Canada, but unfortunately these are based on growth rates and have only very short sample lengths.

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guarantee that the macroeconomic variables identified in the cointegrating vector are in fact linked to the future revenue stream of the asset or to future discount rates. Chart 5: Alternative Measures of Asset Price Misalignments

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5. Empirical Results 5.1 Linear Model Results The regression coefficients, p-values and R2 for the 16 linear models estimated are reported in Annex 2 (Tables A.2.1 and A.2.2). The RMSEs and confusion rates in the out-of-sample forecast exercise are reported in Annex 4 (Tables A.4. 1 and A.4.2). Charts of forecast and actual values are at the end of the paper (Charts X and Y). GDP growth: At the 1 and 4 -quarter horizons, we observe that the presence of financial variables is of little value in forecasting GDP growth. The RMSEs are marginally below those of the base-case models. More substantial improvements occur at the 8 and 12-quarter horizons. For example, the simple inclusion of the growth rate of new house prices (Model 11) lowers the RMSE of BM2 from 1.86 to 0.84 at the 8-quarter horizon. It appears that growth measures of asset prices outperform our simple measures of misalignments, but that there is some value to including both. Augmenting BM2 at the 8 -quarter horizon with the growth of stock prices and the stock price gap results in an RMSE of 0.57, compared with 0.84 when the stock price gap is excluded. CPI inflation: (to be added) 5.2 Threshold Model Results The estimated thresholds and p-values for the 16 threshold models estimated are reported in Annex 3 (Tables A.3.1 and A.3.2). The RMSEs and confusion rates in the out-of-sample forecast exercise are reported in Annex 5 (Tables A.5. 1 and A.5.2). Charts of forecast and actual values are at the end of the paper (Charts X and Y). GDP growth: As shown in Annex 3, we are detecting many significant thresholds (p-values under 0.05) in the relationship between GDP and asset prices. They seem to be especially prevalent at the 4-quarter horizon, as 17 of the 20 models have p-values under 0.05, thereby indicating that the relationship between asset prices and GDP growth over this horizon may be somewhat non-linear. This should be of some interest to policy-makers, as the four-quarter horizon for GDP growth is given serious consideration in policy decisions. Several significant thresholds are also uncovered at the 8 and 16-quarter horizons especially.

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Given that many significant thresholds are being uncovered at the 4-quarter horizon, it should come as little surprise that most of the improvements in forecasting performance relative to the linear models are being uncovered at this horizon as well. For example, for model 10, which regresses output growth against the output gap, term spread and filtered stock price gap, we find that the root mean squared error falls by nearly 1.00, indicating that the average forecast error each quarter is about a ful l percentage lower. The best forecasting model at this horizon is model 15 with a threshold effect attached to the stock price gap, where the root mean squared error is 0.65. As can be seen in Figure 7, this model tracks the movements in GDP growth very well over the 2000-2004 period. The significant thresholds in equations 14 and 15 also yield some notable forecast improvements at the 8-quarter horizon. We find that the RMSE drop from 1.24 to 0.36 in equation 14 when the real house price growth is used as a threshold variable, and from 1.03 to 0.39 in equation 15 when the stock price gap is the threshold variable. The corresponding graphs in Figure 8 reveal that these two models have tracked the 8 -quarter growth rate very tightly over the last five years. At the longer horizons we find that the use of threshold models adds little to forecast performance relative to the linear models; in fact, the forecast performance is often worsened. CPI inflation: (To be added) 6. Concluding Remarks Overall, we find that stock and housing prices can be of use in predicting output growth. The marginal improvements in forecast performance tend to increase as the forecast horizon expands, indicating that asset prices are best viewed as medium -run indicators of economic performance. Our findings also show that non -linearities in the relationships between asset prices and output growth are most prevalent at the 4 and 8-quarter forecasting horizons, indicating that users of asset prices as indicators of output growth could exploit such nonlinearities to improve their forecasts. These findings will be subject to a series of robustness checks, including the choice of filtering techniques applied, alternative measures of asset price gaps, and tests of misspecification and tests of model stability.

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Gropp, Reint, Jukka Vesala and Giuseppe Vulpes (2002) Equity and Bond Market Signals as Leading Indicators of Bank Fragility European Central Bank Working Paper 150

Gurkaynak, Refet S. (2005) Econometric Tests Asset Price Bubbles: Taking Stock Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board , Washington, D.C.

Helbling, T. and M. Terrones (2003) Real and Financial Effects of Bursting Asset Price Bubbles When Bubbles Burst IMF World Economic Outlook April.

Helbling, Thomas and Marco Terrones (2003) When Bubbles Burst World Economic Outlook Chapter II 61-94.

Jagric, Timotej (2003) Forecasting with Leading Economic Indicators – A Non-Linear Approach Prague Economic Papers.

Knox, Thomas, James H. Stock and Mark W. Watson (2001) Empirical Bayes Forecasts of One Time Series Using Many Predictors National Bureau of Economics Research Technical Working Paper 269.

16

Kiyotaki, Nobuhiro and John Moore (1997) Credit Cycles Journal of Political Economy, 1997, vol. 105, no. 2).

Krystalogianni, Alexandra, George Matysiak and Sotiris Tsolacos (2004) Forecasting UK Commercial Real Estate Cycle Phases with Leading Indicators: A Probit Approach.

Leeper, Eric M. and Jennifer E. Roush (2003) Putting ‘M’ Back in Monetary Policy National Bureau of Economics Research Working Paper 9552.

Lunde, Asger and Allan G. Timmermann (2003) Duration Dependence in Stock Prices: An Analysis of Bull and Bear Markets Centre for Economic Policy Research no. 4104.

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Sarantis, Nicholas (2001) Nonlinearities , Cyclical Behaviour and Predictability in Stock Markets: International Evidence International Journal of Forecasting 17 (2001) 459-482.

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Tetlow, Robert J. (2003) Monetary Policy, Asset Prices and Misspecification The Robust Approach to Bubbles with Model Uncertainty Board of Governors of the Federal Reserve System.

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17

Annex 1: Gauthier and Li (2004) BEAM measure of equity price misalignments (more to be added here) Gauthier and Li (2003) define equity price misalignments as the error-correction term from the cointegration relationship linking the equity market to its long-run determinants. Such a measure indicates how far equity prices are from their fundamental value but does not tell us which of the equity prices or their fundamentals or both would move to close this gap. In BEAM, the permanent components of every variables is estimated in the VECM (including stock prices) using the identification methodology suggested in King, Plosser, Stock and Watson (1989). This allows the construction of a stock market gap defined as the difference between stock prices and its permanent component. The gap is therefore the transitory component of stock market which, by definition should not last. This way, a negative (positive) gap suggests that stock prices are expected to increase (decrease) in the absence of further shocks.

18

Annex 2: In-sample Linear Regression Results

Table A.2.1: GDP Growth (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2 BaseCase 1 1. AR1 0.555

(0.01) 0.31 0.169

(0.26) 0.02 -0.096

(0.59) 0.00 -0.263

(0.11) 0.06 -0.473

(0.02) 0.21

BaseCase 2 2. AR1 Output Gap (QPM) Term Spread

0.514 (0.00) -0.273 (0.11) 0.348 (0.02)

0.38 0.101 (0.45) 0.001) (0.01) 0.834 (0.00)

0.40 0.000 (1.00) -0.002 (0.00) 0.629 (0.00)

0.43 -0.070 (0.65) -0.000 (0.72) 0.485 (0.00)

0.39 -0.227 (0.38) 0.000 (0.35) 0.312 (0.01)

0.33

Alternatives Relative to BaseCase 1 3. AR1 TSX (real growth)

0.509 (0.00) 0.017 (0.12)

0.33 0.065 (0.71) 0.034 (0.07)

0.07 -0.260 (0.17) 0.059 (0.04)

0.09 -0.588 (0.02) 0.154 (0.01)

0.30 -0.837 (0.01) 0.146 (0.04)

0.33

4. AR1 TSX Gap (BEAM)

0.503 (0.00) 0.043 (0.03)

0.33 0.107 (0.52) 0.047 (0.07)

0.06

-0.162 (0.36) 0.039 (0.12)

0.04 -0.328 (0.06) 0.023 (0.34)

0.07 -0.528 (0.01) 0.015 (0.50)

0.22

5. AR1 TSX Gap (Filter)

0.525 (0.00) 2.247 (0.36)

0.31 0.212 (0.21) -2.297 (0.48)

0.02 -0.045 (0.81) -2.592 (0.29)

0.01 -0.183 (0.25) -1.913 (0.10)

0.05 -0.382 (0.11) -0.937 (0.27)

0.17

6. AR1

New House Prices (real growth)

0.507 (0.00) 0.049 (0.47)

0.31 0.314 (0.09) -0.098 (0.31)

0.08 0.272 (0.23) -0.247 (0.02)

0.30 0.324 (0.00) -0.360 (0.00)

0.72 0.031 (0.82) -0.314 (0.00)

0.68

7. AR1 New House Price Gap (Filter)

0.482 (0.00) -0.271 (0.01)

0.36 0.092 (0.55) -0.490 (0.00)

0.39 -0.110 (0.59) -0.414 (0.00)

0.51 -0.190 (0.26) -0.307 (0.00)

0.49 -0.345 (0.13) -0.187 (0.00)

0.39

Alternatives Relative to BaseCase 2 8. AR1

Output Gap (QPM) Term Spread TSX (real growth)

0.480 (0.00) -0.230 (0.17) 0.339 (0.03) 0.012 (0.25)

0.38 0.117 (0.47) -0.086 (0.09) 0.661 (0.00) 0.018 (0.19)

0.44 0.823 (0.00) -1.078 (0.00) 0.308 (0.01) 0.006 (0.68)

0.70 0.687 (0.00) -0.882 (0.00) 0.327 (0.00) 0.046 (0.06)

0.78 -0.185 (0.60) -0.263 (0.06) 0.287 (0.01) 0.077 (0.23)

0.49

19

Table A.2.1: GDP Growth (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2 9. AR1

Output Gap (QPM) Term Spread TSX Gap (BEAM)

0.359 (0.01) -0.185 (0.25) 0.593 (0.00) 0.071 (0.00)

-0.067 (0.64) -0.002 (0.00) 0.630 (0.00) 0.039 (0.05)

0.006 (0.97) 0.001 (0.00) 0..898 (0.00) 0.067 (0.00)

0.51 0.47 -0.079 (0.56) -0.00 (0.72) 0.483 (0.00) 0.003 (0.88)

0.38 -0.161 (0.52) 0.000 (0.32) 0.330 (0.00) -0.012 (0.54)

0.33

10. AR1 Output Gap (QPM) Term Spread TSX Gap (Filter)

0.435 (0.00) -0.313 (0.06) 0.415 (0.01) 5.290 (0.03)

0.40 0.085 (0.61) 0.001 (0.04) 0.843 (0.00) 0.815 (0.79)

0.40 0.017 (0.92) -0.002 (0.00) 0.625 (0.00) -0.926 (0.62)

0.42 -0.016 (0.91) 0.000 (0.91) 0.490 (0.00) -0.948 (0.12)

0.40 -0.164 (0.57) 0.001 (0.14) 0.324 (0.01) -1.623 (0.14)

034

11. AR1 Output Gap (QPM) Term Spread


0.446 (0.00) -0.491 (0.01) 0.251 (0.09) 0.127 (0.07)

0.41 0.526 (0.00) -1.044 (0.00) 0.233 (0.19) 0.129 (0.09)

0.53 0.858 (0.00) -1.091 (0.00) 0.343 (0.01) 0.011 (0.86)

0.71

0.875 (0.00) -0.714 (0.00) 0.157 (0.00) -0.227 (0.00)

0.93 0.379 (0.03) -0.372 (0.00) -0.051 (0.69) -0.307 (0.00)

0.79


New House Price Gap (Filter)

0.502 (0.00) -0.243 (0.17) 0.218 (0.21) -0.145 (0.27)

0.38 0.143 (0.32) -0.078 (0.10) 0.410 (0.04) -0.271 (0.02)

0.49 -0.018 (0.93) -0.059 (0.05) 0.211 (0.24) -0.291 (0.00)

0.57

-0.136 (0.45) 0.001 (0.97) 0.205 (0.18) -0.229 (0.00)

0.51 -0.296 (0.30) 0.013 (0.27) 0.151 (0.33) -0.141 (0.01)

0.39

Combinations

20

Table A.2.1: GDP Growth (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2 13. AR1 Output Gap (QPM) Term Spread

TSX (real growth) TSX Gap (BEAM)

0.342 (0.01) -0.160 (0.31) 0.576 (0.00) 0.008 (0.39) 0.068 (0.00)

0.44 0.065 (0.67) -0.058 (0.21) 0.808 (0.00) -0.004 (0.77) 0.066 (0.00)

0.51 0.860 (0.00) -1.101 (0.00) 0.366 (0.01) -.021 (0.126) 0.041 (0.00)

0.75

0.747 (0.00) -0.948 (0.00) 0.277 (0.00) 0.063 (0.01) -0.022 (0.12)

0.79 -0.036 (0.90) -0.374 (0.02) 0.225 (0.06) 0.12 (0.08) -0.044 (0.00)

0.79


New House Prices (real growth) New House Price Gap (Filter)

0.431 (0.00) -0.463 (0.21 0.112 (0.52) 0.129 (0.046) -0152 (0.231)

0.42 0.449 (0.32) -1.032 (0.00) 0.019 (0.89) 0.170 (0.02) -0.243 (0.00)

0.59 0.707 (0.00) -1.033 (0.00) 0.191 (0.15) 0.077 (0.24) -0.215 (0.02)

0.76

0.900 (0.00) -0.727 (0.00) 0.614 (0.00) -0.235 (0.00) 0.019 (0.47)

0.92 0.695 (0.00) -0.548 (0.00) -0.028 (0.67) -0.456 (0.00) 0.212 (0.00)

0.86


New House Prices (real growth) New House Prices Gap (filter) TSX (real growth) TSX Gap (filter)

0.371 (0.00) -0.450 (0.02) 0.173 (0.34) 0.112 (0.08) -0.150 (0.22) 0.004 (0.72) 3.899 (0.09)

0.42 0.464 (0.00) -1.045 (0.00) 0.018 (0.91) 0.171 (0.02) -0.243 (0.00) -0.003 (0.88) -0.122 (0.97)

0.58 0.572 (0.02) -0.950 (0.00) 0.191 (0.16) 0.083 (0.11) -0.238 (0.01) 0.019 (0.42) 1.465 (0.45)

0.77

0.702 (0.00) -0.651 (0.00) 0.122 (0.00) -0.226 (0.00) 0.004 (0.84) 0.038 (0.16) -0.043 (0.96)

0.94 0.559 (0.00) -0.502 (0.00) -0.035 (0.62) -0.426 (0.00) 0.200 (0.007) 0.037 (0.00) -2.186 (0.09)

0.88


21

Table A.2.1: GDP Growth (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2

New House Prices (real growth) TSX (real growth)

22

Table A.2.2: CPI All Items (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2 BaseCase 1 1. AR1

0.560 (0.00)

0.34 0.661 (0.00)

0.64 0.631 0.00

0.57 0.600 0.00

0.50 0.615 0.00

0.511

BaseCase 2 2. AR1 Output Gap (QPM)

0.555 0.00 0.187 0.09

0.35 0.619 0.00 0.004 0.00

0.67 0.462 0.00 0.004 0.00

0.637 0.374 0.00 0.003 0.00

0.55 0.352 0.00 0.002 0.00

0.55


0.544 0.00 -0.007 0.35

0.34 0.627 0.00 0.001 0.91

0.61 0.430 0.00 0.026 0.17

0.43 0.507 0.00 0.092 0.00

0.45 0.705 0.00 0.166 0.00

0.71


0.562 0.00 0.003 0.85

0.33 0.674 0.00 0.018 0.19

0.64 0.664 0.00 0.036 0.06

0.59 0.651 0.00 0.042 0.03

0.52 0.673 0.00 0.040 0.03

0.53


0.55 0.00 -1.242 0.52

0.34 0.671 0.00 3.279 0.06

0.66 0.623 0.00 5.024 0.04

0.62 0.508 0.00 4.621 0.02

0.56 0.457 0.00 3.952 0.02

0.57

6. AR1


0.570 0.00 0.013 0.79

0.33 0.557 0.00 0.115 0.00

0.52 0.535 0.00 0.178 0.00

0.71 0.423 0.01 0.143 0.01

0.47 0.374 0.09 0.048 0.36

0.25

7. AR1


0.514 0.00 0.144 0.04

0.36 0.471 0.00 0.118 0.00

0.53 0.423 0.00 0.123 0.11

0.60 0.361 0.00 0.059 0.39

0.49 0.353 0.00 0.007 0.89

0.49

Alternatives Relative to BaseCase 2 8. AR1 Output Gap (QPM) TSX (real growth)

0.544 0.00 0.173 0.11 -0.006 0.47

0.35 0.514 0.00 0.127 0.00 -0.006 0.44

0.70 0.537 0.00 0.488 0.00 0.013 0.29

0.81 0.517 0.00 0.436 0.00 0.046 0.00

0.76 0.613 0.00 0.211 0.00 0.123 0.00

0.77

9. AR1 Output Gap (QPM) TSX Gap (BEAM)

0.555 0.00 0.187 0.09 -0.000 1.00

0.34 0.631 0.00 0.003 0.00 0.015 0.24

0.67 0.492 0.00 0.004 0.00 0.029 0.07

0.66 0.427 0.00 0.003 0.00 0.040 0.02

0.61 0.422 0.00 0.002 0.00 0.047 0.01

0.65

10. AR1 Output Gap (QPM)

0.539 0.00 0.234

0.35 0.632 0.00 0.003

0.68 0.485 0.00 0.003

0.67 0.401 0.00 0.003

0.58 0.377 0.00 0.002

0.59

23

Table A.2.2: CPI All Items (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2 TSX Gap (Filter)

0.06 -2.543 0.26

0.00 2.619 0.09

0.00 3.308 0.05

0.00 3.163 0.02

0.00 2.862 0.03



0.529 0.00 0.241 0.05 -0.030 0.57

0.35 0.547 0.00 0.272 0.02 0.043 0.26

0.56 0.549 0.00 0.361 0.00 0.071 0.05

0.81 0.428 0.00 0.482 0.00 0.009 0.84

0.72

0.396 0.00 0.428 0.00 -0.050 0.27

0.59



0.519 0.00 0.137 0.23 0.119 0.11

0.36 0.490 0.00 0.111 0.00 0.113 0.10

0.72 0.385 0.00 0.113 0.00 0.062 0.39

0.72 0.333 0.00 0.103 0.00 -0.002 0.98

0.62 0.327 0.00 0.086 0.01 -0.045 0.38

0.61

Combinations 13. AR1 Output Gap (QPM)


0.545 (0.00) 0.171 (0.13) -0.006 (0.46) 0.002 (0.89)

0.34 0.516 (0.00) 0.128 (0.00) -0.011 (0.17) 0.018 (0.212)

0.71 0.529 (0.00) 0.481 (0.00) 0.005 (0.73) 0.014 (0.127)

0.82

0.495 (0.00) 0.436 (0.00) 0.027 (0.04) 0.019 (0.10)

0.78 0.588 (0.00) 0.224 (0.00) 0.100 (0.00) 0.013 (0.161)

0.78



0.500 (0.00) 0.180 (0.18) -0.023 (0.66) 0.114 (0.13)

0.35 0.526 (0.00) 0.264 (0.03) 0.035 (0.39) 0.047 (0.48)

0.56 0.563 (0.00) 0.339 (0.00) 0.102 (0.00) -0.064 (0.05)

0.82

0.431 (0.00) 0.446 (0.00) 0.097 (0.14) -0.149 (0.01)

0.78 0.343 (0.00) 0.442 (0.00) 0.083 (0.23) -0.190 (0.00)

0.70


New House Prices (real growth) New House Prices Gap (filter) TSX (real growth)

0.498 (0.00) 0.165 (0.21) -0.021 (0.70) 0.110 (0.70) -0.009 (0.61)

0.34 0.512 (0.00) 0.269 (0.03) 0.029 (0.54) 0.047 (0.48) -0.006 (0.46)

0.56 0.588 (0.00) 0.313 (0.00) 0.012 (0.00) -0.064 (0.04) 0.017 (0.18)

0.82

0.484 (0.00) 0.422 (0.01) 0.072 (0.34) -0.118 (0.05) 0.016 (0.61)

0.80 0.541 (0.00) -0.301 (0.00) 0.033 (0.56) -0.106 (0.08) 0.068 (0.00)

0.80

24

Table A.2.2: CPI All Items (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 ai R 2 ai R 2 ai R 2 ai R 2 ai R 2

TSX Gap (BEAM)

0.004 (0.81)

0.010 (0.57)

-0.002 (0.84)

0.015 (0.37)

0.018 (0.05)



25

Annex 3: Estimated Thresholds

Table A.3.1: GDP Growth (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value BaseCase 1 1. AR1 5.59 0.911 1.84 0.542 4.07 0.590 4.69 0.665 1.47 0.632


0.037

0.382

-0.013

0.000

1.813

0.062

2.030

0.014

1.853

0.001


-19.16

0.003

13.59

0.032

-2.01

0.008

13.23

0.075

22.80

0.740


-3.40

0.132

4.18

0.000

5.85

0.004

2.213

0.126

9.089

0.119


-0.022

0.578

0.028

0.007

-0.080

0.359

-0.001

0.053

-0.030

0.582

6. AR1


2.903

0.189

0.451

0.678

1.931

0.148

0.051

0.076

-0.678

0.043


-0.261

0.004

0.963

0.000

1.005

0.947

-1.535

0.182

-1.588

0.307



-19.16

0.034

13.59

0.050

-0.708

0.062

11.16

0.318

22.80

0.035

9. AR1

Output Gap (QPM)

26


value τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value Term Spread TSX Gap (BEAM)

1.264

0.145

-5.88

0.003

3.103

0.000

-1.700

0.225

1.011

0.000


-0.011

0.309

0.029

0.029

-0.080

0.359

-0.001

0.455

0.062

0.452



2.903

0.379

2.457

0.005

1.931

0.158

0.051

0.041

-0.425

0.012



-1.349

0.015

-1.773

0.000

-0.289

0.014

-1.534

0.001

-1.588

0.940

Combinations 13. AR1 Output Gap (QPM) Term Spread


-19.16 1.011

0.032 0.083

1.736 5.159

0.010 0.006

-0.708 5.852

0.139 0.001

10.88 -1.700

0.135 0.236

12.36 1.011

0.043 0.001


27


value τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value Term Spread


-4.702 -1.349

0.351 0.023

-1.368 -1.987

0.001 0.000

-1.01 -0.913

0.000 0.000

0.051 -1.535

0.017 0.003

-5.471 -0.615

0.006 0.663


New Hou se Prices (real growth) New House Prices Gap (filter) TSX (real growth) TSX Gap (BEAM)

-1.588 1.012

0.090 0.338

-1.773 1.087

0.000 0.000

-1.443 5.852

0.000 0.000

-1.535 8.046

0.009 0.193

1.005 1.012

0.485 0.018



-0.425 9.777

0.147 0.099

2.457 1.736

0.016 0.053

0.843 -0.708

0.000 0.005

0.051 13.23

0.115 0.284

-0.425 22.80

0.032 0.055

28

Table A.3.2: CPI All Items (1981: Q4 – 2004: Q4) Regressors k=1 k=4 k=8 k=12 k=16 τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value BaseCase 1 1. AR1

2.507 0.826 3.535 0.000 3.708 0.000 3.111 0.034 5.476 0.203

BaseCase 2 2. AR1 Output Gap (QPM)

0.198

0.045

0.835

0.107

0.958

0.147

-3.509

0.187

0.576

0.005


-24.04

0.078

6.050

0.109

-4.261

0.006

17.30

0.134

10.88

0.853


-3.892

0.109

-0.486

0.002

5.159

0.877

-3.453

0.715

4.417

0.218


-0.044

0.795

0.061

0.075

0.048

0.298

-0.049

0.590

0.048

0.884

6. AR1


-4.921

0.000

-3.550

0.216

-2.096

0.002

-0.678

0.116

2.487

0.057

7. AR1


1.005

0.001

-1.401

0.038

-1.986

0.001

-2.095

0.026

-2.245

0.836

Alternatives Relative to BaseCase 2 8. AR1 Output Gap (QPM) TSX (real growth)

-24.04

0.080

6.050

0.028

-0.241

0.000

9.916

0.119

18.39

0.136

9. AR1 Output Gap (QPM) TSX Gap (BEAM)

-3.892

0.066

-0.486

0.021

-2.354

0.814

9.089

0.709

5.159

0.022

10. AR1

29


value τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value Output Gap (QPM) TSX Gap (Filter)

-0.053

0.654

0.061

0.019

0.048

0.019

-0.032

0.257

0.048

0.633



-4.921

0.000

-2.294

0.015

-2.096

0.023

1.860

0.087

2.487

0.007



1.005

0.004

-1.092

0.114

-1.588

0.000

-1.401

0.000

-1.588

0.010

Combinations 13. AR1 Output Gap (QPM) Term Spread


-24.04 -3.892

0.256 0.034

9.121 1.012

0.040 0.003

-0.708 -6.401

0.006 0.388

17.29 9.380

0.326 0.609

18.39 5.159

0.270 0.046



-4.921 1.005

0.001 0.002

-2.294 -1.092

0.002 0.000

-0.812 -1.637

0.009 0.000

-3.077 -1.401

0.142 0.001

0.320 -1.588

0.055 0.003


30


value τ̂ p-

value τ̂ p-

value τ̂ p-

value τ̂ p-

value Term Spread

New House Prices (real growth) New House Prices Gap (filter) TSX (real growth) TSX Gap (BEAM)

1.005 -3.892

0.001 0.032

-1.086 2.095

0.001 0.000

-1.637 1.011

0.000 0.001

-1.401 9.090

0.000 0.096

-1.588 4.417

0.002 0.009



-4.921 -21.80

0.000 0.039

-2.294 -9.525

0.002 0.053

-0.812 7.600

0.006 0.013

0.320 13.50

0.233 0.419

2.903 22.80

0.000 0.319

31

Annex 4: Linear Models, Out-of-Sample Forecast Performance

Table A.4.1: GDP Growth (2000:Q1 to 2004:Q4) Regressors k=1 k=4 k=8 k=12 k=16 RMSE C.R. RMSE C.R. RMSE C.R. RMSE C.R. RMSE C.R.

BaseCase 1 1. AR1 1.64 0.79 1.49 1.0 1.58 0.53 1.69 0.32 1.78 0.16 BaseCase 2 2. AR1 Output Gap (QPM) Term Spread

1.59 0.78 1.92 0.42 1.86 0.58 1.15 0.26 1.43 0.21


1.56 0.74 2.11 0.58 1.41 0.53 1.00 0.32 1.68 0.37


1.88 0.63 1.83 0.74 1.64 0.58 1.90 0.37 2.10 0.21


1.78 0.68 1.52 0.74 1.59 0.21 1.63 0.26 1.78 0.16

6. AR1 New House Prices (real growth)

1.64 0.79 1.51 0.79 1.17 0.63 1.03 0.68 1.35 0.26


1.64 0.79 1.49 0.84 1.40 0.63 1.23 0.47 1.48 0.21

Alternatives Relative to BaseCase 2 8. AR1 Output Gap (QPM) Term Spread TSX (real growth)

1.57 0.79 1.86 0.58 0.84 0.26 0.68 0.21 1.35 0.53


Term Spread TSX Gap (BEAM)

1.67 0.63 1.78 0.53 1.74 0.63 1.56 0.32 1.88 0.53


1.63 0.68 2.07 0.42 1.91 0.47 1.21 0.26 1.48 0.21



1.56 0.68 1.41 0.32 0.84 0.31 0.61 0.21 1.32 0.21

12. AR1 Output Gap (QPM) Term Spread New House Price Gap (Filter)

1.57 0.79 1.67 0.42 1.53 0.53 1.22 0.32 1.71 0.26


TSX (real growth)

1.62 0.63 1.82 0.47 0.57 0.37 1.03 0.26 1.66 0.47

32


TSX Gap (BEAM) 14. AR1 Output Gap (QPM) Term Spread


1.51 0.68 1.27 0.42 0.87 0.32 0.63 0.21 1.17 0.21



1.55 0.63 1.49 0.63 1.24 0.58 0.70 0.37 1.26 0.26



1.53 0.68 1.45 0.58 1.03 0.37 0.57 0.21 1.39 0.47

33

Annex 5: Threshold Models, Out-of-Sample Forecast Performance


BaseCase 1 1. AR1 1.62 0.74 1.63 0.47 1.18 0.63 2.04 0.47 1.51 0.63


1.48 0.58 1.03 0.32 0.83 0.37 1.46 0.58 1.61 0.53


1.84 0.63 1.56 0.58 1.31 0.58 1.28 0.53 1.59 0.63


1.74 0.53 1.41 0.47 1.56 0.47 1.02 0.74 1.25 0.58


1.57 0.58 1.68 0.53 1.09 0.53 1.37 0.63 1.35 0.79

6. AR1 New House Prices (real growth)

1.63 0.74 1.47 0.74 1.02 0.63 0.90 0.47 1.55 0.58


1.84 0.68 1.51 0.74 1.17 0.42 1.64 0.52 0.82 0.32



1.65 0.63 1.40 0.53 1.14 0.47 1.38 0.47 1.67 0.42


Term Spread TSX Gap (BEAM)

1.57 0.58 1.02 0.42 1.18 0.53 0.78 0.42 1.94 0.37


1.69 0.47 1.09 0.32 0.93 0.32 1.22 0.42 1.07 0.42



1.52 0.74 1.47 0.58 0.56 0.21 0.97 0.47 2.10 0.58



1.38 0.74 1.01 0.10 0.67 0.21 1.74 0.53 0.87 0.53


TSX (real growth)

1.49 1.49

0.53 0.37

1.35 0.96

0.37 0.32

1.15 1.06

0.47 0.63

1.48 1.24

0.42 0.37

1.55 1.97

0.32 0.37

34


TSX Gap (BEAM) 14. AR1 Output Gap (QPM) Term Spread


1.52 1.24

0.63 0.68

1.04 1.09

0.37 0.21

0.36 0.57

0.37 0.42

0.95 1.49

0.42 0.58

1.38 1.40

0.47 0.47



1.45 1.47

0.58 0.32

1.00 0.65

0.16 0.21

0.49 0.39

0.42 0.32

1.69 1.55

0.37 0.42

1.32 1.95

0.53 0.42



1.39 1.52

0.53 0.47

1.36 1.41

0.42 0.32

0.62 0.80

0.32 0.42

1.21 1.45

0.42 0.42

2.00 1.80

0.53 0.53

forecasting with bubblesweb.hec.ca/scse/articles/tkacz.pdf · 2005-05-10 · forecasting with...

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