Forecasting the Brazilian Yield Curve Using
Forward-Looking Variables
Fausto Vieira
Sao Paulo School of Economics Fundação Getulio Vargas
Fernando Chague
Department of Economics University of São Paulo
Marcelo Fernandes
Sao Paulo School of Economics Fundação Getulio Vargas
Abstract: This paper proposes a simple model using a factor-augmented VAR
methodology with Nelson-Siegel factors to forecast the Brazilian term structure of interest rates. We extract the principal components for the FAVAR from weekly macroeconomic and financial indicators, many of which are forward-looking. Our model entails a significant improvement on the short-term forecasting relative to the extant models in the literature. For longer horizons, our methodology also fares well, yielding small forecast errors. For instance, the mean absolute forecast errors are 30-40% lower than the random walk benchmark after two months horizon. The out-of-sample analysis shows that including forward-looking indicators is the key to improve the predictive ability of the model.
JEL classification: E58, C38, E47
1. Introduction
The yield curve of treasury bonds plays a central role in pricing financial assets and in shaping market expectations. As such, accurately estimating and forecasting the yield curve is of great importance for the Treasury, central bankers and market participants in general. Usually, macro-finance models are able to outperform the random walk benchmark only at horizons longer than six months1. In stark contrast, we show that our model improves on the random walk benchmark even at the one-month horizon. In this paper, we propose a forecasting strategy for the yield curve that improves the standard benchmark model at short- and medium-horizons.
We proceed in two steps. First, we estimate the Nelson-Siegel three-factor model to pin down the evolution of the level, slope and curvature of the yield curve. We then regress these factors on a comprehensive data set of macroeconomic and finance variables using a Factor Augmented VAR (FAVAR) model. At last, we employ the autoregressive structure of the FAVAR model to predict the future path of the yields at each maturity. Our forecasts of the yield curve beat the random walk benchmark as early as at the one-month horizon. This represents a significant improvement to the available models that produce only meaningful at the six-month horizon onwards. At the one-month horizon, our model forecasts errors are, on average, 5 to 10% lower than those of the random walk, whereas it improves by 30 to 40% on the random walk for longer horizons.
Our data set includes a number of forward-looking macroeconomic variables that heavily contribute to the forecasting improvements. For example, the market expectations of the Brazilian GDP growth in the next year and up to five years ahead are important to predicting yield curve movements. The expected government balance and debt to GDP ratio also play a significant role.
We use the Nelson-Siegel parametrization to extract the yield curve from bond prices. The Nelson-Siegel parametrization decomposes the movements of the entire yield curve into level, slope and curvature changes. In order to predict the future yield curve, we follow Diebold and Li (2006) in forecasting the future values of the level, slope and curvature factors without imposing no-arbitrage restrictions.
To deal with the large number of leading indicators, we use the FAVAR econometric model. The estimation is in two steps. We first extract the principal components from almost 200 real-time weekly series since March 2007. In the second step, we estimate
1
a FAVAR model, using the level, slope and curvature factors that summarize the term structure and a few of the principal components we obtain from the first step.
Our forecasting strategy relates to a set of papers that uses economics variables to explain the yield curve. Dielbod and Li (2006) use a vector autoregressive for estimating and forecasting the three factors, with good results at the medium-term horizons. Ang and Piazzesi (2003) impose a no-arbitrage hypothesis in the estimation of their VAR model with macroeconomic variables. Moench (2008) combines no-arbitrage affine models with the FAVAR methodology of Bernanke and Boivin (2003) and Bernanke, Boivin and Eliaz (2005) by employing a large number of economic series to explain future changes in the US term structure. As a result, Moench is able to outperform the forecast of the extant models in the literature at intermediate horizons (6 and 12 months ahead). Faria and Almeida (2014) employ Moench’s (2008) affine FAVAR specification to predict the Brazilian yield curve. They find a similar result in that the model predicts best at horizons longer than 6 months. For shorter terms (say, 1 month and 3 months), the random walk exhibits lower forecast errors.
In contrast to Moench, we do not impose no-arbitrage restrictions. Duffee (2011) evaluates the importance of the latter when using the term structure to forecast future bond yields. He shows that it is much more important to assume that the level of the term structure is close to a random walk than imposing the absence of arbitrage opportunities. Joslin et al (2011) confirm that the forecasting ability of no-arbitrage models is at par with that of unconstrained VAR models. See also Carriero (2011) and Carriero and Giacomini (2011) for more details on how no-arbitrage restrictions affect the prediction of the term structure of interest rates.
Our main contributions to this literature are as follows. In contrast to Diebold and Li (2006) and Moench (2008), we are able to improve the forecast of the yield curve even at the short term. Our forecasting model is simple. It augments Diebold and Li’s (2006) VAR for the level, slope and curvature factors of the term structure, with the principal components we extract from the economic data series as in Moench (2008) plus the target interest rate of the Brazilian Central Bank (SELIC). The main difference is on the nature of these economic drivers. We not only employ data at a higher frequency than the literature (weekly rather than monthly), but also focus on forward-looking indicators to have a more timely assessment of the current financial and economic situation. We organize the remainder of this paper as follows: Section 2 briefly reviews the Nelson-Siegel approach to the term structure of interest rates and then describes the forecast model we employ. Section 3 provides an in-depth description of the data,
whereas Section 4 discusses both in- and out-of-sample results. Section 5 examines the reasons why our forecast model outperforms the extant models in the literature. Section 6 concludes.
2. The forecast model
We combine different methods in the literature to come up with a simple forecast model for the Brazilian yield curve. In particular, we predict the future values of the level, slope and curvature factors of the Nelson-Siegel decomposition of the yield curve using a FAVAR framework that incorporates the information conveyed by the principal components of a wide array of macroeconomic and financial variables. In some sense, it combines Diebold and Li’s (2006) VAR model for the level, slope and curvature factors with Stock and Watson’s (2002b) idea of conditioning on a small number of principal components extracted from a large number of macroeconomic and market indicators. This is very similar to what Moench (2008) and Huse (2011) do. The main difference with respect to Moench is that we do not impose no-arbitrage conditions given that they may hurt the forecasting performance of the model (see, for instance, Carriero, 2011; Carriero and Giacomini, 2011). In contrast to Huse’s model, we allow for shocks in the level, slope and curvature factors through a VAR specification as in Diebold and Li (2006).
The Nelson-Siegel decomposition of the yield curve posits that the yield at maturity n depends on (1)
where the betas respectively reflect the level, slope and curvature of the term structure. To broadly incorporate the information about the economy in the VAR model for the betas, we must consider a large number of macroeconomic and financial indicators. For this task, we take the advantage of the availability of a rich data set with 199 variables, including 68 forward-looking macroeconomic variables. However, we cannot just add these variables as exogenous regressors in the VAR specification because it would lead to an excessive number of parameters to estimate. Stock and Watson (2002b) circumvent this issue by constructing a small number of diffusion indices that proxy for the broad economic conditions by means of a principal component analysis applied to a large number of macroeconomic indicators.2 They show that this leads to
2
Although the principal component analysis formally requires independent and identically distributed observations, Stock and Watson (2002a) and Doz, Giannone and Reichlin (2012) show that it performs
much better forecasts that alternative VAR models for US macroeconomic time series. We follow their lead and carry out a principal component analysis to a wide array of macroeconomic and financial time series in order to reduce the dimensionality of our FAVAR model.
The Nelson-Siegel factors and the k x 1 vector of principal components augmented by the target interest rate of Brazilian Central Bank are the main ingredients of our forecasting model for the yield curve. Let denote a (k + 3) vector of constants, a (k + 3) x (k + 3) matrix of order-p lag polynomials, and a vector of reduced form shocks, then the FAVAR reads
(2) .
Bernanke, Boivin and Eliasz (2005) suggest two manners to estimate a FAVAR: Two-steps (principal components plus VAR estimation) or a Bayesian method based on Gibbs sampling. They show that both methods produce similar results, though the two-step estimation not only is computationally simpler, but also yields results that are more plausible. As such, for the sake of convenience, we proceed with the estimation in two steps. We first extract the level, slope and curvature factors of the yield curve as well as the k principal components relating to the macroeconomic and financial conditions, and then estimate
(3)
Finally, we use the estimates of the above FAVAR model to forecast the yield curve h-weeks ahead given the future values of the level, slope and curvature factors with with set information in t. To predict the yield curve, we forecast in equation (4) and then back out the expected yields by means of
(4)
.
In view that equations (3) and (4) essentially describes a factor-augmented version of the VAR model that Diebold and Li (2006) propose for the level, slope and curvature factors of the Nelson-Siegel decomposition of the yield curve, we will henceforth refer to the above model by DL-FAVAR.
In the next section, we describe the data we employ to proxy for the macroeconomic and market conditions in the economy.
similarly to full maximum likelihood estimation for a large panel in the context of both static and dynamic factor models, respectively.
3. Data description 3.1 Macroeconomic data
There are many high-frequency economic indicators in Brazil since 1980s because of the high inflationary process that governed prices until mid-1990s. At the time, there was a large demand for weekly price indices to enable fairer readjustments of contracts. A second wave of higher frequency economic data dates back to the implementation of the inflation targeting in late 1990s by the Central Bank of Brazil. To monitor market expectations, the latter weekly releases the Focus report, with market forecasts of daily indicators of activity, inflation, external and fiscal accounts for current month or year until projections for next 5 years ahead. This set of high frequency indicators has relevant forward-looking information about the Brazilian economy.
Altogether, this means the Brazilian data provide lots of useful high-frequency information about future movements in the yield curve. In particular, we focus on a data set with almost 200 weekly indicators from March 2007 to December 2014. Note that we entertain data only as from the first week of March 2007 because this is the first month for which FGV has computed its daily consumer price index. In addition, the Brazilian Treasury started issuing bonds with longer maturities of Brazilian by the end of 2006, with liquidity picking up only in 2007.
We entertain a multitude of data sources. Inflation-related variables computed from commodity, producer and consumer price indices constitute 42% of the database. They relate to price changes in the last month as well as expected variation in the current month or in a determined period (e.g. next 12 months or in 5 years). Producer prices are from CEASA, a distribution center for crops, fruits and vegetables, and other cooperatives. We gather commodity prices from Bloomberg, whereas we collect consumer prices at the weekly frequency from FIPE (São Paulo only) and from FGV at the daily frequency. All expectations data come from the weekly Focus report that the Central Bank of Brazil releases every Monday. Apart from mean and median forecasts, the Focus database also includes information about the standard deviations of the short- and medium-term forecasts of inflation, activity, fiscal and balance payments series. They amount to the second largest group of data, with a share of 17% of the overall database.
The share of real activity and fiscal series are respectively 15% and 5% of the database. They are mainly from the Central Bank of Brazil, except to a couple of daily activity indicators (e.g., electric energy consumption and car sales). All indicators released by the Central bank of Brazil (namely, real activity, external accounts and
fiscal data) concern market expectations over a certain horizon, e.g. current month, next year, or next 5 years. Altogether, over 40% of the inflation, real activity and fiscal time series we consider are forward looking indicators, thereof providing a more timely information about the Brazilian outlook in the short and long term.
Finally, we extract financial and risk indicators from Bloomberg. They correspond to 11% and 7% of the database, respectively. They include real-time indicators of the Brazilian economy, such as the 5-year Brazil CDS, the local stock market index, and the currency contracts outstanding, as well as of the global economy, such as the US financial index, Latin America EMBI and the fed funds rate.
To extract principal components from this broad range of variables, we first make sure that every time series is stationary by taking first differences, if necessary. We construct diffusion indices in two different manners. First, we extract the first 5 principal components of the full set of indicators in the database. Table 1 displays the variables that have the highest correlations with each principal component. The first component explains more than 40% of the overall variation and correlates mostly with Bloomberg’s economic surprise index3 and with the Brazilian stock market index. The second relates chiefly to real activity outlook and expected inflation over the next 5 years. The third and fourth components reflect primarily fiscal indicators and forecasting uncertainty, respectively. Finally, the fifth principal component seems to convey information mostly about emerging markets in general.
As an alternative, we also consider extracting principal components from data subsets in order to improve on interpretability. In the first subgroup, we extract the first principal component of the inflation-related variables. It explains 60% of the variation within this category, correlating mostly with food inflation in consumer and producer prices. The second group gathers real activity indicators, including the economic surprise index, daily current activity indicators, and growth expectation. The third set of variables is about fiscal indicators, ranging from market expectations of the future debt and of the budget surplus to the values of the Brazilian EMBI and CDS. The remainder categories relate to risk, forecast uncertainty and financial indicators. It is interesting to observe that their first principal components explain over 80% of the within variation.
3.2 The level, slope and curvature of the Brazilian yield curve
3
This index gauges economic news by the average difference between the expected and actual values of some economic indicators.
Differently from the US Treasury emissions, the Brazilian Treasury issues bonds with a specific expiration date. For example, in January 2014, the Treasury issued a fixed rate bond with a maturity of 11 years, expiring in January 2025 (NTN-F 25). It is very likely that the Brazilian Treasury will launch another 11-year bond (NTN-F 27) only in January 2016. This feature of the Brazilian term structure of government bonds makes the Nelson-Siegel decomposition particularly interesting for it allows us to back out a fixed maturity yield curve. We estimate equation (1) in two different manners. We first estimate every parameter, that is to say, the level, slope and curvature factors given by the betas as well as the parameter λ that governs at which term of the yield curve curvature is strongest. The nonlinear least square estimation indicates a mean value of 0.017 for the latter. Next, we re-estimate the model fixing λ to the value at which the mean absolute difference between the actual and estimated yields is smallest. This yields a much higher value for λ at 0.266. It is interesting to note that Diebold and Li (2006) use λ = 0.0609 to fit the term structure in the US.
Despite the very different values for λ, both methods yield very similar fits to the Brazilian yield curve. As such, we opt for simplicity and proceed using a fixed λ of 0.266. Figure 1 provides a graphical comparison between these estimates using the term structure estimated by ANBIMA as a benchmark.4 The average yield curve is indeed very similar from 2009 to 2014, regardless of the method (fixing λ or not, adding a second curvature factor or not). Figure 2 complements the comparison by plotting the 1-year and 10-year yields as predicted by the Nelson-Siegel model with fixed λ and their ANBIMA counterparts.
4. Empirical Results 4.1 Preliminary Evidence
Bernanke and Boivin (2003) show that central bankers consider a wide range of data to make decisions about interest rates. They show that dimension-reduction techniques, such as Stock and Watson’s (2002b) diffusion indices, typically improve the forecast of economy and inflation indicators, with clear benefits to the estimation of the central bank’s reaction function.
We adapt Bernanke and Boivin’s (2003) augmented Taylor rule to the weekly frequency as follows:
(5)
4
ANBIMA is the Brazilian Association of Banks and Financial Institutions. They publish since 2009 yield curve estimates assuming the Svensson (1994) model, which includes an additional curvature factor to the Nelson-Siegel specification.
where so as to explicitly link the target interest rate of the Central Bank of Brazil (namely, the SELIC rate) to the diffusion indices (namely, the
principal components from a set of indicators). Table 3 reports the results for the SELIC regression on the overall principal components as well as on the first principal component of each category. The principal components are jointly significant, even if the loading on the real activity and forecast uncertainty components are not statistically different from zero. As expected, the estimates not surprisingly indicate that more inflation and deterioration of fiscal indicators lead to a higher interest rate.
Next, to estimate (5), we proxy the inflation and growth gaps by the difference between the expected inflation and GDP growth over the next 12 months and the expected inflation and GDP growth in the long run (as measured by market expectations over the next 5 years in the Focus database). Table 4 shows that the information on the principal components is indeed useful for the policymaker decision. Table 5 shows that there is a strong correlation between the target interest rate and short-term yields, though it decreases with both the forecast horizon and the maturity. The correlation of the overall principal components with the term structure is surprisingly low for every horizon, except to the principal component reflecting information about emerging markets at the 6- and 12-month horizons. In stark contrast, the first principal component of each category commands strong correlation with the yield curve. The real activity and forecast uncertainty components have lower predictive ability, though are definitely important at longer horizons. The opposite applies for the financial and risk indicators, who boast strong influence only in the short run. The fiscal-related component significantly affects the yield curve at every horizon, whereas the inflation-related component features surprisingly little correlation with the yield curve.
4.2 Fitting the yield curve
In this section, we report the estimation results of the DL-FAVAR model for the level, slope and curvature of the Brazilian yield curve. As before, we entertain specifications with the overall principal components in Table 1 as well as with the first principal components of each category in Table 2. We find that the FAVAR models do a very good job in fitting the Nelson-Siegel factors. Figure 3 depicts fitted and actual values for different maturities, illustrating the fact that fitting well the level, slope and curvature factors indeed suffices to fit each yield individually. The largest estimation error is of circa 80 bps, occurring in the last quarter of 2008.
Table 6 reports the sample average and standard deviation of the actual and fitted yields, as well as of the magnitude of the estimation error. The most striking finding is
that the mean absolute errors are not only small at about 20 bps, but also very stable across maturities. This is in stark contrast to the findings in Moench (2008) and in Faria and Almeida (2014). Their mean absolute errors increase with the maturity, reaching 90 to 150 bps for the 6- and 12-month yields.
Altogether, we find that the FAVAR captures well both the short and long ends of the Brazilian term structure of interest rates. In the next section, we examine whether the good in-sample performance also translates into superior forecasts.
4.3 Out-of-sample analysis
In this section, we assess the forecasting performance of our DL-FAVAR model relative to the extant models in the literature. As aforementioned, we consider a number of variations according to the sort of principal component we employ to augment the VAR model. The first version is DL-FAVAR(all), which considers the first 5 overall principal components. The second variant is DL-FAVAR(group), which employs the first principal component of each of the 6 subgroups we contemplate. Finally, DL-FAVAR(fira) restricts attention to the first principal components of the subgroups relating to fiscal, inflation and real activity indicators. Note that we include the target interest rate in every model in order to account for the monetary dimension.
Apart from the usual random walk (RW) benchmark, we also contemplate the forecasting performance of a standard autoregressive model (AR), Diebold and Li’s (2006) VAR model for the level, slope and curvature factors (DL), and Moench’s (2008) affine FAVAR using the overall principal components as driving factors for the short rate (Moench). We expect the random walk to be a challenging benchmark, especially at shorter forecasting horizons (see Joslin et al., 2011). We assume no drift and hence it is not necessary to estimate any parameter in RW. In all other instances, we choose the lag structure using Akaike’s information criterion.
The affine FAVAR model employs the overall principal components as driving factors for the short rate. In particular, we assumes as in Moench (2008) that
(6) ,
where denotes the short rate and is a vector of white noises with covariance matrix . After estimating the parameters in (6), we impose no-arbitrage considerations by minimizing the market prices of risk ( , ) in
,
for some initial conditions ( ). Next, we obtain the future values of the n-year zero rate using the affine nature of the model: .
We estimate every model using data from the first week of March 2007 to the last week of May 2012, and then assess forecasting performance for the remainder 137 weeks up to the last week of December 2014. Although we fix the lag structure to the one we choose in sample, we re-estimate each model to produce the corresponding iterated forecasts at different horizons as we expand the estimation window up to December 2014.5
Table 7 reports the mean absolute forecast error we obtain for each model across different maturities and horizon. The DL-FAVAR models for the level, slope and curvature factors of the yield curve perform very well, improving forecasts by 10 to 25 bps at the 3-month horizon and by 30 to 40 bps at the 6-month horizon. In particular, DL-FAVAR(fira) has the best performance for 1- and 2-month-ahead forecasts irrespective of the maturity of the yield. This suggests that fiscal, inflation and real activity indicators are key to explain the short-run movements in the yield curve. Interestingly, it actually fares very well at any horizon, especially for the yields with shorter maturity. Although DL-FAVAR(all) entails the smallest mean absolute forecast errors for the longer maturities, focusing exclusively on fiscal, inflation and real activity indicators definitely pays off for the 1- and 3-year maturities at any horizon. In contrast, DL-FAVAR(group) does not shine as the other DL-FAVAR models, even if it nevertheless compares well with the RW, DL and Moench alternatives.
These results are very promising. Diebold and Li (2006) and Moench (2008) show that their models provide better forecasts for the US yield curve than the random walk benchmark only at longer horizons, say, 6 months or more. We indeed find a very similar pattern for the Brazilian yield curve. DL and Moench outperform the RW and AR benchmarks only at the 6-month horizon or longer. This suggests that the success of the DL-FAVAR models is genuine and not due to some peculiarity of the Brazilian term structure of interest rates.
As for the relative size of these refinements, the DL-FAVAR models entail mean absolute forecast errors 5 to 15% smaller than the RW benchmark at the 1-month horizon. The improvement in mean absolute forecast error is even larger for longer
5
See Marcellino, Stock and Watson (2006) for an excellent discussion about the relative advantages and drawbacks of direct and iterated AR forecasts.
horizons, in the range of 30 to 40%. Figure 4 displays box plots for the 3- and 6-month ahead forecast errors for the 1- and 10-year yields. The first row indicates that the forecast error of the DL-FAVAR(fira) for the 1-year yield and of the DL-FAVAR(all) for the 10-year yield have not only the smallest median values at the 3-month horizon, but also the smallest first and third quartiles. A similar pattern also arises for the 6-month ahead forecasts, confirming the apparent superior performance of these models. The one million dollar question is whether these improvements are indeed statistically significant. To answer this question, we run White’s (2000) reality check to put the outstanding performance of the DL-FAVAR models into perspective. The idea is to contemplate alternative scenarios for the forecasting exercise by means of artificial (block) bootstrap samples. In particular, for each bootstrap sample, we compute the relative forecast errors of each model with respect to the benchmark and then back out their empirical distributions by aggregating across the bootstrap samples. The null hypothesis is that the best forecasting model has no predictive superiority over the RW benchmark. Table 8 shows the p-value of White’s reality check for each model.
We find no model that delivers superior 1-week ahead forecasts relative to the RW benchmark. However, the picture changes fast as the forecast horizon increases. At the 1-month horizon, the DL-FAVAR(fira) is the only model for which we reject the null hypothesis of no predictive superiority over the RW for the 1-, 3- and 5-year yields. For horizons longer than 2 months, every DL-FAVAR model beats the RW in a statistically significant manner. At the 10% level, DL also ameliorates the RW forecasts for horizons longer than 3 months if we restrict attention to yields with longer maturities as well as for short-term yields at the 6-month horizon. Finally, we fail to uncover at the usual significance levels any statistical evidence of superior forecasting performance for the AR and Moench models, regardless of the yield maturity and forecast horizon.
5. Where does the superior performance of the DL-FAVAR model come from?
Affine models of the term structure of interest rates typically fail to beat the random walk benchmark at shorter forecast horizons, even if conditioning on macroeconomic variables as in Ang and Piazzesi (2003) and Moench (2008). Even at longer horizons, they do not offer much better forecasts then Diebold and Li’s (2006) VAR model for the level, slope and curvature factors. There is some evidence that macro-finance models render poor predictions because no-arbitrage restrictions are too stringent, hindering forecast performance (Carriero, 2011; Carriero and Giacomini, 2011). Duffee (2011) indeed shows that unrestricted VAR models predict better than arbitrage-free term structure models.
On the other hand, one would expect changes in the macroeconomic and market conditions to drive the movements in the yield curve, especially in the short run. This is why we deem very reassuring that the DL-FAVAR models furnish better forecasts than simpler AR-type models (including the particular case of the random walk) even at shorter horizons. This improvement may stem from the use of higher frequency data and/or of real-time forward-looking indicators, and hence we investigate in the sequel from where it comes.
As for sampling frequency, the existing models in the literature employ monthly rather than weekly data. It is thus natural to ask how much we gain from considering a higher frequency. Table 9 shows how the forecast performance of Diebold and Li’s VAR model for the level, slope and curvature factors changes as we vary the sampling frequency from daily to monthly. Surprisingly, increasing the sampling frequency seems to ameliorate the forecast performance at longer horizons. Indeed, the daily DL model performs poorer than the weekly and monthly models at the 1-month horizon, whereas it slightly beat them at the 3- and 6-month horizons. This pattern is common to every yield, regardless of the maturity. At the 1-month horizon, using weekly data helps predict shorter-maturity yields, whereas the monthly model works best for longer-maturity yields.
We next check whether the DL-FAVAR yield better forecasts because of the forward-looking nature of the economic and financial indicators we employ. Most models in the literature use activity indicators that do not represent the current economic outlook due to late data releases (e.g., activity indicators are typically two months late). In contrast, we exploit several forward-looking indicators based on market expectations. To assess the extra mileage we get from these variables, we estimate weekly DL-FAVAR models using either lagged or forward-looking economic indicators and then assess their forecast performance against the weekly DL model. Table 9 shows that adding forward-looking economic indicators pays off, improving on DL forecasts by 5 to 20% at the 1-month horizon and by 25 to 40% for longer horizons. The same does not apply to the DL-FAVAR model based exclusively on lagged economic indicators, whose relative performance is quite poor for every maturity and horizon.
Altogether, we conclude that the key is in the forward-looking nature of the factors with which we augment the VAR model for the level, slope and curvature of the yield curve. Increasing the sampling frequency indeed seems to contribute very little to the success of the DL-FAVAR model.
6. Conclusion
This paper proposes to forecast future values of yields at different maturities by means of a FAVAR model for the level, slope and curvature of the yield curve. In particular, we estimate a VAR model for a system that includes not only the Nelson-Siegel factors of the Brazilian yield curve, but also the principal components of a large number of macroeconomic and financial (local or global) indicators. We show that our forecasting approach outperforms the extant models in the literature, including the random walk benchmark, even at shorter horizons. Further analysis reveals that using forward-looking state variables is vital to produce better forecasts.
We defer the assessment of external validity to future research. In particular, we will examine whether we observe similar forecast improvements using US data. There is no reason to believe our findings automatically carry through. First, it is perhaps the case that the term structure of interest rates in Brazil has a very particular dynamics. Second, it is surprisingly easier to gather a larger number of forward-looking indicators in Brazil than in the US. This may hinder the predictive ability of the DL-FAVAR model given that market expectations about the economic and financial outlooks are very informative.
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Table 1
The overall principal components
Principal Component Analysis (all series) correlation
Factor 1 - 40.9% of total variance
G10 economic data surprise 0.873
Emerging market economic data surprise 0.807
US economic data surprise 0.786
Latin America economic data surprise 0.612
Ibovespa index 0.528
Factor 2 - 23.7% of total variance
CPI expected in 5 years -0.355
Import average growth in next 3 to 5 years 0.335 GDP growth expectation in next 12 months 0.329
Import growth in 5 years 0.324
Industrial production growth in next 12 months 0.311 Factor 3 - 14.8% of total variance
Daily electricity consumption in the South States 0.256
Crops commodity index (in log) 0.239
Brazilian daily electric energy consumption 0.213 Debt as percentage of GDP in 5 years -0.201 Fiscal balance forecast in next 3 to 5 years 0.197 Factor 4 - 8.7% of total variance
Standard deviation of service GDP forecast in next 12 months 0.195 Daily electric energy consumption in the Northeast States 0.195 Debt as percentage of GDP in next 12 months -0.176 Standard deviation of debt (%GDP) forecast in next 3 to 5 years 0.175
Energy commodity index (in log) -0.156
Factor 5 - 4.9% of total variance
EMBI ex Brazil -0.563
Asia financial condition index 0.550
EMBI Latin America -0.538
Import growth in next 12 months 0.490
Export growth in next 12 months 0.488
This table lists the variables with the highest correlation with each of the first 5 principal components of the panel of macroeconomic and financial variables.
Table 2
The first principal component of each subgroup
Principal Component Analysis by Subgroup Correlation
Inflation Factor - 60.2% of the variation in inflation-related series
Daily tomatoes price 0.980
Vegetable and fruit producer price 0.901
Vegetable and fruit consumer price 0.511
Food producer price 0.491
Consumer price expected for the current month 0.398 Real Activity Factor - 43.3% of the variation in activity-related series
G10 economic data surprise 0.855
Import growth in 5 years 0.476
Export growth in next 12 months 0.402
GDP growth expectation in next 12 months 0.331 Industrial production growth expectation in next 12 months 0.290 Fiscal Factor - 78.9% of the variation in fiscal-related series
Debt (%GDP) forecast in next 3 to 5 years 0.990 Primary fiscal balance forecast in 5 years 0.482 Fiscal balance forecast in next 3 to 5 years -0.395
CDS Brazil -0.288
Primary fiscal balance forecast in 12 months 0.277 Risk Factor - 96.3% of the variation in financial risk-related series
VIX 0.998
BRL 1-month risk reversal 0.893
EMBI Brazil 0.782
CDS Latin America 0.773
EMBI ex Brazil 0.693
Forecast Uncertainty Factor - 80.1% of the variation in forecast standard deviations
Standard deviation of import growth forecast in 5 years 0.987 Standard deviation of export growth forecast in next 3 to 5 years 0.960 Standard dev of industrial production growth forecast in next 3 to 5 years -0.620 Standard deviation of balance of payment in next 3 to 5 years 0.638 Standard deviation of industrial production growth forecast in 5 years -0.371 Financial Factor - 84.4% of all financial indicators variance
US financial conditions -0.917
Eurozone financial conditions -0.907
US 5-year breakeven inflation -0.806
TED spread 0.574
Open interest in Ibovespa futures -0.569
This table displays the correlations of the first principal component extracted from the different subgroups (real activity, inflation, fiscal, financial risk, forecasting uncertainty and financial series) on individual variables for each set. We list the five variables with the highest correlation with the first principal component of each subgroup.
Table 3
Policy rules based on factors
PCA(all) PCA(group) C 10.3685 C 10.3685 (0.1944) (0.1461) PCA 1 -0.0082 Inflation 0.3656 (0.0030) (0.1125) PCA 2 0.0078 Activity 0.0051 (0.0040) (0.0056) PCA 3 -0.0086 Fiscal 0.2797 (0.0049) (0.0755) PCA 4 0.0037 Uncertainty -1.4074 (0.0085) (0.3690) PCA 5 -0.0046 Financial 0.0018 (0.0089) (0.0117) Risk 1.4127 (0.3566) R-square 0.128 R-square 0.492 This table documents factor-based rules for the target interest rate of the Central bank of Brazil. In the first rule, we regress the target interest rate on the first 5 principal components of the full panel of macroeconomic and financial variables. In the second rule, we regress the target interest rate on the first principal component of each subgroup. The numbers in parentheses are estimated parameter standard deviations.
Table 4
Augmented Taylor rules
(A) (B) (C)
Past target interest rate 0.987 0.971 0.905
(0.003) (0.017) (0.020)
CPI forecast for the next 12 months 26.044 11.362 3.834
(4.633) (6.964) (0.762)
GDP growth forecast for the next 12 months 10.832 4.716 1.193
(3.063) (2.857) (0.388)
Predicted target interest rate based on PCA(all) 0.576
(0.282)
Predicted target interest rate based on PCA(group) 0.844
(0.037)
R-square 0.966 0.967 0.970 This table reports the regression results for equation (5). Column (A) displays the coefficient estimates for the traditional Taylor rule, whereas columns (B) and (C) show the estimates for augmented Taylor rules that include the target interest rate predicted by the factor models in Table 3. The numbers in parentheses are estimated parameter standard deviations.
Table 5
Correlation between yields and principal components
Yield Selic PCA 1 PCA 2 PCA 3 PCA 4 PCA 5 Inflation Activity Fiscal Risk Uncertainty Financial
Contemporaneus correlation 1 year 0.93 -0.23 0.29 -0.14 -0.02 0.01 -0.02 0.23 0.49 0.46 -0.33 0.47 3 years 0.79 -0.21 0.35 -0.11 -0.04 -0.07 0.03 0.19 0.49 0.53 -0.33 0.54 5 years 0.71 -0.20 0.37 -0.10 -0.03 -0.13 0.05 0.18 0.47 0.57 -0.31 0.58 7 years 0.66 -0.20 0.37 -0.09 -0.03 -0.17 0.06 0.17 0.45 0.58 -0.30 0.59 10 years 0.62 -0.19 0.37 -0.08 -0.03 -0.19 0.07 0.17 0.43 0.59 -0.29 0.60
Correlation with 1-month lagged factors
1 year 0.87 -0.16 0.26 -0.11 -0.06 0.12 0.00 0.23 0.53 0.39 -0.37 0.40
3 years 0.75 -0.12 0.30 -0.08 -0.08 0.04 0.04 0.22 0.55 0.50 -0.38 0.50
5 years 0.68 -0.12 0.31 -0.07 -0.08 -0.03 0.05 0.22 0.52 0.56 -0.36 0.57 7 years 0.64 -0.12 0.31 -0.07 -0.08 -0.07 0.06 0.21 0.50 0.59 -0.34 0.60 10 years 0.61 -0.12 0.30 -0.06 -0.08 -0.10 0.06 0.21 0.48 0.61 -0.33 0.61
Correlation with 3-month lagged factors
1 year 0.70 0.01 0.19 -0.06 -0.11 0.33 0.02 0.27 0.61 0.13 -0.50 0.14
3 years 0.61 0.03 0.13 -0.04 -0.13 0.24 0.02 0.34 0.62 0.22 -0.53 0.22
5 years 0.58 0.02 0.10 -0.04 -0.13 0.16 0.00 0.36 0.60 0.30 -0.51 0.30
7 years 0.57 0.01 0.08 -0.04 -0.13 0.11 0.00 0.37 0.57 0.33 -0.49 0.34
10 years 0.55 0.00 0.07 -0.04 -0.13 0.08 -0.01 0.38 0.55 0.36 -0.47 0.37
Correlation with 6-month lagged factors
1 year 0.40 0.14 0.18 -0.01 -0.08 0.52 0.07 0.35 0.66 -0.08 -0.69 -0.08
3 years 0.39 0.09 0.10 0.03 -0.07 0.44 0.07 0.47 0.64 0.05 -0.74 0.05
5 years 0.43 0.04 0.05 0.04 -0.06 0.37 0.06 0.52 0.60 0.14 -0.72 0.15
7 years 0.44 0.01 0.02 0.05 -0.05 0.32 0.06 0.53 0.58 0.19 -0.70 0.19
10 years 0.45 -0.01 0.00 0.06 -0.05 0.29 0.06 0.54 0.55 0.23 -0.69 0.23
Correlation with 12-month lagged factors
1 year -0.07 0.20 -0.02 0.04 0.04 0.47 0.03 0.57 0.48 -0.15 -0.79 -0.16
3 years 0.05 0.04 -0.07 0.02 0.03 0.39 0.00 0.68 0.42 0.00 -0.81 0.00
5 years 0.13 -0.04 -0.08 0.01 0.02 0.37 -0.02 0.70 0.39 0.07 -0.80 0.08
7 years 0.17 -0.08 -0.08 0.00 0.01 0.35 -0.03 0.71 0.37 0.11 -0.78 0.11
10 years 0.20 -0.12 -0.08 0.00 0.00 0.34 -0.03 0.71 0.35 0.14 -0.76 0.14
This table reports the correlation between yields and the principal components of the overall panel of macroeconomic and financial variables as well as of their subgroups.
Table 6
Average and standard deviation of the actual and fitted yields
Yield Mean Standard Deviation
Actual Fitted AE Actual Fitted AE
ALL 1 year 0.1086 0.1085 0.0015 0.0187 0.0187 0.0056 3 years 0.1165 0.1165 0.0019 0.0168 0.0166 0.006 5 years 0.1195 0.1195 0.0021 0.016 0.0157 0.0062 7 years 0.1208 0.1208 0.0022 0.0157 0.0155 0.0063 10 years 0.1219 0.1219 0.0023 0.0157 0.0154 0.0063 GROU P 1 year 0.1086 0.1086 0.0015 0.0187 0.0187 0.0055 3 years 0.1165 0.1165 0.0018 0.0168 0.0166 0.006 5 years 0.1195 0.1195 0.002 0.016 0.0158 0.0061 7 years 0.1208 0.1208 0.0022 0.0157 0.0155 0.0062 10 years 0.1219 0.1219 0.0022 0.0157 0.0154 0.0063 This table reports sample mean and standard deviation of the actual and DL-FAVAR fitted yields as well as of the corresponding absolute errors (AE) for each maturity.
Table 7
Mean absolute forecast errors across maturities and horizons Yield DL-FAVAR (all) DL-FAVAR (group) DL-FAVAR (fira) Diebold- Li Moench RW AR 1 month ahead 1 year 0.315 0.260 0.251 0.309 0.570 0.305 0.307 3 years 0.361 0.352 0.305 0.360 0.749 0.356 0.361 5 years 0.367 0.396 0.336 0.369 0.803 0.366 0.372 7 years 0.380 0.420 0.357 0.379 1.059 0.377 0.385 10 years 0.392 0.442 0.376 0.391 1.096 0.389 0.398 3 months ahead 1 year 0.480 0.471 0.423 0.676 0.895 0.692 0.666 3 years 0.460 0.522 0.458 0.736 0.923 0.722 0.704 5 years 0.464 0.558 0.505 0.734 0.945 0.742 0.727 7 years 0.480 0.585 0.543 0.741 1.122 0.759 0.744 10 years 0.496 0.612 0.578 0.752 1.174 0.774 0.760 6 months ahead 1 year 1.024 0.869 0.819 0.966 1.348 1.149 1.128 3 years 0.906 0.903 0.810 0.990 1.232 1.218 1.203 5 years 0.836 0.904 0.837 1.004 1.239 1.266 1.253 7 years 0.822 0.908 0.863 1.020 1.302 1.291 1.280 10 years 0.828 0.914 0.892 1.038 1.309 1.311 1.300 12 months ahead 1 year 1.646 1.672 1.436 1.525 2.248 2.325 2.300 3 years 1.335 1.491 1.288 1.519 1.789 2.136 2.124 5 years 1.231 1.470 1.319 1.488 1.740 2.044 2.037 7 years 1.212 1.481 1.399 1.475 1.485 2.007 2.001 10 years 1.207 1.496 1.474 1.488 1.399 1.990 1.984 This table summarizes the mean absolute forecast error of each model. We estimate every model using data from March 2007 to April 2012 and then produce h-month ahead forecasts, with h = 1, 3, 6 and 12, for the period running from May 2012 to December 2014. DL-FAVAR(all) refers to the DL-FAVAR model with the first 5 principal components of the overall panel of macroeconomic and financial variables. DL-FAVAR(group) is the DL-FAVAR model with the first principal component of each subgroup (inflation, activity, fiscal, forecasting standard deviation, financial risk and financial indicators), whereas DL-FAVAR(fira) employs only the first principal components from the fiscal, inflation, and real activity subgroups of macroeconomic and financial variables.
Table 8
White’s reality check results Yield DL-FAVAR (all) DL-FAVAR (group) DL-FAVAR (fira) Diebold-Li Moench AR 1 week ahead 1 year 0.923 0.810 0.130 0.910 1.000 0.147 3 years 0.867 0.870 0.137 0.483 1.000 0.157 5 years 0.857 0.933 0.200 0.447 1.000 0.167 7 years 0.907 0.923 0.837 0.146 1.000 0.203 10 years 0.965 0.973 0.900 0.927 1.000 0.903 1 month ahead 1 year 0.903 0.033 0.013 0.963 0.997 0.867 3 years 0.993 0.997 0.016 0.997 1.000 0.986 5 years 1.000 0.993 0.037 1.000 1.000 0.993 7 years 0.807 0.980 0.080 0.927 1.000 0.863 10 years 0.837 0.864 0.110 0.867 1.000 0.906 2 months ahead 1 year 0.012 0.000 0.000 0.913 0.990 0.137 3 years 0.000 0.000 0.000 0.950 1.000 0.120 5 years 0.000 0.013 0.000 0.946 0.997 0.163 7 years 0.014 0.053 0.000 0.873 0.990 0.173 10 years 0.014 0.005 0.017 0.867 1.000 0.196 3 months ahead 1 year 0.000 0.000 0.000 0.140 0.986 0.097 3 years 0.000 0.000 0.000 0.120 0.980 0.133 5 years 0.000 0.000 0.000 0.103 0.993 0.140 7 years 0.000 0.000 0.000 0.080 0.967 0.156 10 years 0.000 0.000 0.000 0.097 0.990 0.127 6 months ahead 1 year 0.083 0.007 0.000 0.093 0.970 0.122 3 years 0.010 0.017 0.000 0.126 0.836 0.923 5 years 0.000 0.000 0.000 0.060 0.097 0.146 7 years 0.000 0.000 0.000 0.067 0.853 0.103 10 years 0.000 0.000 0.000 0.040 0.167 0.117
This table reports the p-values of White’s reality check. We compare the out-of-sample forecasting performance of each model relative to the random walk benchmark. The null hypothesis is that the mean squared forecast errors of each model is equal to the mean squared forecast error of the random walk.
Table 9
Forecasting performance for different frequencies and data sets Frequency analysis Database analysis Yield DL Daily DL Weekly DL Monthly DL-FAVAR (nfl) DL-FAVAR (fira)
1 month ahead 1 year 0.336 0.309 0.353 0.308 0.251 3 years 0.387 0.360 0.376 0.412 0.305 5 years 0.398 0.369 0.368 0.450 0.336 7 years 0.410 0.379 0.369 0.471 0.357 10 years 0.422 0.391 0.371 0.488 0.376 3 months ahead 1 year 0.670 0.676 0.722 0.637 0.423 3 years 0.694 0.736 0.758 0.789 0.458 5 years 0.685 0.734 0.735 0.826 0.505 7 years 0.691 0.741 0.735 0.846 0.543 10 years 0.711 0.752 0.750 0.863 0.578 6 months ahead 1 year 0.948 0.966 1.081 1.052 0.819 3 years 0.965 0.990 0.977 1.115 0.810 5 years 0.987 1.004 0.970 1.144 0.837 7 years 1.005 1.020 1.007 1.148 0.863 10 years 1.023 1.038 1.035 1.152 0.892
This table reports the mean absolute forecast errors of the DL model using daily, weekly and monthly data as well as of the DL-FAVAR(nfl) model that uses no forward-looking indicator. For ease of comparison, we report in the Iast column the results of our preferred model DL-FAVAR(fira).
Figure 1
This figure shows the average yield curve implied by the Nelson-Siegel model with fixed and time-varying λ for a sample ranging only from 2009 to 2014. We restrict attention to data as from 2009 in order to match the period for which ANBIMA publishes their yield curve based on the Svensson model.
Figure 2
This figure compares the 1- and 10-year yields implied by the Nelson-Siegel model with fixed λ with the yields released by ANBIMA. Note that the latter series start only in 2009.
Figure 3
This figure plots actual and fitted yields for different maturities
Figure 4
This figure displays box plots for the 3- and 6-month forecast errors for the 1- and 10-year yields.
3-month ahead forecasts of the 1-year yield 3-month ahead forecasts of the 10-year yield
6-month ahead forecasts of the 1-year yield 6-month ahead forecasts of the 10-year yield
.000 .005 .010 .015 .020 .025 MO ENC H RW AU TO_R EGR S DIEB OLD _LI DL_F AV AR_AL L DL_F AVA R_G ROUP DL_F AV AR_F IRA .000 .004 .008 .012 .016 .020 .024 .028 .032 MO ENC H RW AU TO_ RE GR S DIE BOLD _LI DL_FAV AR _ALL DL_FAV AR _GR OU P DL_FAV AR _FIRA .000 .005 .010 .015 .020 .025 .030 .035 MO EN CH RW AU TO_R EGR S DIEB OL D_L I DL_F AV AR_AL L DL_F AVA R_G ROUP DL_F AV AR_F IRA .000 .004 .008 .012 .016 .020 .024 .028 .032 .036 MO EN CH RW AU TO_R EGR S DIEB OLD _LI DL_F AVA R_AL L DL_F AVA R_G ROUP DL_F AV AR _FIR A
