diffusion indexes: the case of...
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Diffusion Indexes:
The Case of Singapore
Lim You Jie, Benedict
May 30, 2010
Abstract
This paper uses factor analysis, a technique that allows researchers to utilize large
cross-sectional macroeconomic data, for the forecasting of inflation and the
production index in Singapore. This paper follows closely the techniques used in
Stock and Watson’s paper on Diffusion Indexes (1998), and utilizes updated
factor selection techniques proposed by Bai and Ng (2001). The factor models
used in this study contains 155 quarterly series observed over the period 1992:Q2
to 2009:Q4. The results based on the RMSE of the 1, 2, 3, 4, and 8 period ahead
forecasts indicate that factor models outperform other conventional forecasting
models such as the random walk model, autoregressive model as well as the
Phillips Curve at the 4 and 8 quarter ahead horizon.
Keywords: Factor Model, AO Random Walk, Diffusion Indexes, Singapore, Forecast Accuracy.
_____________________________________
I am grateful to my thesis advisor Professor Giorgio Primiceri for patiently advising me during the
process of writing this thesis. I am also thankful to Hyerim Shin for her assistance with MATLAB coding.
1. Introduction
Diffusion indexes are averages of the contemporaneous values of a large number of time
series such as the gross domestic product, unemployment rate and money supply. These indexes
have been historically calculated by economists who use their expert opinion to decide what time
series should be included in the indexes and how much weight should be allocated to each series.
These indexes are then used to allow policymakers to identify turning points in the economy
such as economic upswings or recessions. A drawback to this method of constructing diffusion
indexes is that the economist’s beliefs are projected into his or her selection criterion. In this
paper, we use a factor analysis, a technique that does not impose a priori assumptions and
economic theory, to construct diffusion indexes for the Singapore economy.
Accurate forecasts play an integral role in how businesses plan their future expenditures and
financial strategies, how individuals adjust their savings, and also how investors adjust their
expectations for stock markets. For policy makers, central banks need accurate inflation forecasts
to allow them to analyze the potential ramifications of their monetary policies, and also to allow
them to analyze what inflation would look like without any governmental intervention. Accurate
forecasts of the Industrial Production Index (IPI) are also important because it is the most
reliable quantitative indicator on economic activity, allowing economists to better analyze
business cycles.
Figure 1.
Figure 2.
From Figures 1 and 2 graphs the quarterly annualized inflation and IPI growth
respectively. We see that Singapore generally has had low and stable inflation, while IPI growth
has been very volatile, ranging from 40% to negative 48% growth.
The ability to achieve accurate forecasts in Singapore is even more pressing because of
its status as a country with a small open economy. As an island state without natural resources,
Singapore is heavily dependent on trade, as seen by the fact that the sum of merchandise imports
and exports is 362% as a ratio of GDP (World Bank, World Development Indicators). Thus it is
not surprising that the Monetary Authority of Singapore (MAS) follows an exchange-rate centric
monetary policy, and has helped Singapore maintain low inflation rates and sustained economic
growth in the last 45 years of independence. The exchange rate is pegged to an undisclosed
trade-weighted basket of currencies, and adjustments to the exchange rate are dependent mainly
on expected inflationary pressures. Any misjudgments and policy misstep would have serious
repercussions on Singapore’s economy.
The main goal of this paper is to assess if the use of factor models for forecasting actually
improves upon the accuracy of inflation and IPI growth forecasts when compared to other
conventional forecasting models. The other models that are used for comparison are the Atkeson-
Ohanian (2001) random walk model, direct autoregressive model, multivariate leading indicator
model, and the Phillips Curve model. We find that factor models do indeed show a significant
improvement in only inflation forecasting accuracy as compared to the competing models at the
4 and 8 quarter ahead forecasting horizon. Factor models do not improve the forecasting
performance of IPI growth forecasts. Moreover, when using the Diebold-Mariano (1995) and
Giancomini-White (2003) test to test for the statistical significance of this improvement in
forecasting accuracy, we find that this improvement is generally significant only at the 8 quarter
ahead horizon. And lastly, we do not find any conclusive improvement in forecast accuracy
when implementing the Bai-Ng (2002) information criterion in the selection of the optimal
number of factors to be included in the forecasting model.
The paper is structured as such: The second section of the paper surveys the related
existing literature. The third section gives a detailed list of the macroeconomic variables used in
the paper and its associated transformations. Section four presents the theoretical factor model
and the algorithms used to derive the common factors, as well as the methodology used to
determine the optimal number of common factors. Section five describes the other forecasting
models. The sixth section gives a brief description of the general forecast methodology. In
section seven we detail and analyze the empirical results from all of our forecasting models, and
we lastly conclude in section eight.
2. Literature Review
Factor models have been used in research since Sargent and Sims (1977) as well as
Geweke (1977) used factor analysis as a way to analyze macroeconomic activity without
imposing assumptions and economic theory a priori. Their main contribution was to discover that
with the use of factor models, a small number of factors could be extracted to account for the
majority of the variation observed in major economic aggregates. Recently, there has been an
increase in the use of factor models in forecasting around the world, spearheaded by James H.
Stock and Mark W. Watson in a series of papers (1998, 1999, 2002) that analyzed the viability of
utilizing factor models for forecasting inflation and IPI in the United States. The recent
renaissance in factor analysis as an approach towards macroeconomic forecasting can be
attributed to breakthroughs such as the advancement of estimation techniques developed by
Stock and Watson (1998), where they used the principal components method to estimate factor
model parameters. Moreover, the availability of a wide variety of macroeconomic data due to
increased use of computers, recent computational advancements that allowed researchers to
process large volumes of data, and the appeal of driving forecasts based on a large number of
variables without any a priori assumptions, all made factor models more accessible to both
researchers and policymakers.
Many economists and policymakers around the world have conducted empirical studies
utilizing factor models for forecasting, for example Gosselin and Tkacz Kotlowski (2008) in the
case of Poland, Gosselin (2001) in the case of Canada, Kunovac (2007) in the case of Croatia,
and also Angelini, Henry and Mestre (2001) in the case of the Euro area. There have not been
many papers written to analyze how factor models can be used in the context of Singapore, but
the most notable would be a paper by Chow and Choy (2009) where the authors utilized a
dynamic factor model to analyze and forecast business cycles in Singapore. This paper differs
from Chow and Choy (2009) in that we used a modified and updated dataset, we took a different
approach in the factor model forecast methodology, and included a larger variety of inflation
forecasting models for comparison so as to better evaluate the effectiveness of factor models in
forecasting.
The resurgence of factor models as a forecasting tool also drove research on how
effective factor models actually are. Stock and Watson (1999a, 2002, 2004) as well as Forni et el
(2001, 2005) compared factor models to other forecast methods and concluded that factor models
did indeed outperform various benchmark forecasting models, especially against small-scale and
simpler forecasting models. However, Groen, Kapetanios and Price (2007), Faust and Wright
(2007), Eickmeier and Ziegler (2006), Schumacher and Dreger (2006) and Gosselin (2001) also
argued that factor models did not necessarily improve forecast performances when compared to
other existing complex large scale data forecasting models, that the major improvement in
forecasting ability of factor models was statistically insignificant. However, the general outlook
on the usefulness of factor models is generally positive, and current research is focused on
improving factor model forecasts in areas such as non-parametric analysis as well as the in the
area of how to determine optimal number of factors.
In this paper, it was imperative to select an appropriate model as a benchmark model
against which factor forecasting models for inflation and IPI could be compared against. The
need for an alternative reliable inflation forecasting model arose when Jaditz and Sayers (1994)
first noted the weakening of inflation forecast capabilities of the Phillips Curve for the United
States since the mid-1980s. Many other similar empirical studies (Stock and Watson 1998, 2008,
Orphanides 2004, Atkeson-Ohanian 2001) also concluded that while the Phillips Curve model
had been a reliable model for policymakers prior the mid-1980s, other forecasting models
performed better than the traditional Phillips Curve during this period. Atkeson-Ohanian (2001)
in particular showed in an extreme case that modern Phillips Curve based models did not
outperform a simple random walk model, a model akin to what they described as predicting
inflation based on a coin-flip.
Further inflation forecasting empirical studies in the case of the United States showed that the
random walk model not only outperformed the Phillips Curve models, but also other more
sophisticated models. In light of the Atkeson-Ohanian (2001) random walk model's strong
performance in inflation forecasting in the United States for the last 20 years, and because the
model does not place any prior assumptions or theory on inflation forecasting, the random-walk
model is an important benchmark model, allowing us to perform an impartial evaluation.
Conceptually, there are no reasons the Atkeson-Ohanian (2001) random walk model cannot be
extended to be used for IPI forecasting. Thus in this paper, we extend use the Atkeson-Ohanian
(2001) random walk model as the main benchmark to compare forecast performance for both
inflation and of the Industrial Production Index.
3. Description of the Data
Since Singapore's independence in 1965, Singapore has maintained an updated,
extensive, and reliable statistics database on the economy and population. The data used in this
study consist mainly of macroeconomic variables specific to the Singapore economy, as well as
data on major macroeconomic variables on foreign countries with strong economic ties to
Singapore. The selection of time-series was closely based on the 132 variables used in the study
by Stock and Watson (2002) for the US economy, as well as the variables Chow and Choy
(2009) utilized in their dynamic factor modeling exercise for the Singapore economy. Departing
from the two papers, we placed an additional emphasis on trade related variables because of the
economy’s heavy dependence on trade as a global trading hub.
Out of the numerous variables initially under consideration, we dropped variables based
on their frequency and the earliest availability of observations so that there would be no
unobserved variables. Monthly data were either aggregated or averaged to obtain data in the
quarterly format depending on the type of time series. The final list of variables selected
consisted of monthly and quarterly data encompassing the period from 1992:2nd
Quarter to
2009:4th
Quarter (71 Observations). In all, 155 variables were used in the study, and the balanced
dataset was then categorized into several macroeconomic groups to facilitate analysis. The 15
macroeconomic groups are: Singapore real GDP and expenditure components, trade indicators,
general price indices, sectorial indicators, labor market variables, construction sector, industrial
production indices, monetary indicators, financial indicators, business expectations, foreign stock
exchange prices, foreign composite leading index, and lastly foreign GDP.
The time series were then transformed with a similar approach that Stock and Watson
(2005) used so that the time series were approximately stationary and thus could be appropriately
used in factor model specification. Generally, the first difference of logarithms was taken for real
quantity variables to get the quarterly percentage change, second difference of logarithms was
taken for price series and monetary indicators, and the first difference was used for nominal
interest rates. Series that were in the form of percentage changes were not adjusted. After the
dataset was transformed, the dataset was adjusted for outliers following Stock and Watson
(2005) by first identifying and then replacing observations with absolute median deviations 6
times larger than the 75th
and 25th
inter quartile range with the median of the previous 5
observations. After de-trending and adjusting the dataset for outliers, the series were then
normalized to have zero mean and unit standard deviation (Stock and Watson 1998). A
comprehensive list of the variables utilized in this paper and applied transformations are listed in
greater detail in Appendix A.
4. The Factor Model
4.1 The Exact Static Factor Model
In this paper, we follow the procedure used by Stock and Watson (1998 and 2002), where
the forecasting is performed using a two-step process. The first step involves estimating the
common factors from the dataset, and the second step involves performing a linear regression of
the inflation rate on the factors. We use the factor model proposed by Stock and Watson (2002)
to estimate the common factors driving most of the variability observed in the data. We assume
that , the vector time series variable that contain useful information for forecasting , can
be represented by:
(1)
where is a (r x 1) common factor, r is the number of common static factors, is a (N x 1)
idiosyncratic disturbance vector, and Λt is the (N x r) factor loading matrix.
To see how the common factors are implemented in this forecasting exercise, we briefly
describe the general forecast model as:
(2)
where is the scalar time series variable to be forecasted, h is the forecast horizon, is the
common factors extracted from (1), is a (m x 1) vector of observed variables (which in this
paper are the lags of ), and is the forecast error.
In this paper, we make the assumptions that eit and ejt for i ≠ j, as well as are mutually
uncorrelated and are i.i.d. We also assume that the factor loadings are constant, where .
All these assumptions combined make the model an exact static factor model and allow us to
estimate the common factors and factor loadings.
4.2 Other Variations of the Factor Model
There are also variations of the factor model that differs in the underlying assumptions
made. One variation is where the idiosyncratic disturbances are allowed to be serially correlated
and cross-sectionally correlated. When these errors are allowed to be weakly correlated, we call
this model an approximate static factor model (Chamberlain and Rothschild 1984, Stock and
Watson 1998).
Another variation is the dynamic factor model which we do not use in this paper is where
we allow lags of the factors to enter the equation, thus we first write the model as:
(3)
(4)
where and are lag polynomials with finite order q, which is the number of dynamic
factors. With appropriate manipulation of equations (3) and (4), Stock and Watson (2002)
showed that the dynamic factor model can be rewritten in the static form of (1) and (2).
(5)
Where ( ) and
4.2 Estimation of the Factors Using Principle Components
Different methods have been proposed to extract the common factors in factor models.
One method proposed by Forni, Hallin, Lippi and Reichlin (2000, 2003), is a two step approach
that finds the solution of a generalized principle component problem. First the factor estimates
are derived from the spectral density matrices of the time series, these estimates are then used to
construct contemporaneous linear combinations of observed data so as to minimize the
idiosyncratic-common variance ratio. In this paper, we use the method of principal components
proposed by Stock and Watson (1998) to extract the common factors.
The static representation of the factor model in (1) allows us, by the method of principal
components, to non-parametrically estimate the factors. We need to solve for the minimum of the
nonlinear least squares objective function of:
( ) ∑ ∑
(6)
Let ( denote the minimizers of (6), and these estimates satisfy the first order
conditions:
∑
∑
(7)
∑
∑
(8)
Solving for the minimum of (6) by substituting (7) or (8) will yield two different
eigenvalue problems, and each problem can be solved by choosing the factors as the
eigenvectors corresponding to the k largest eigenvalues of the matrix ∑
, or the matrix
∑
respectively, where k is the number of factors desired. Thus the algorithm to find the
common factors can be summarized as such: Given a balanced normalized dataset, we find the
matrix of eigenvectors and the corresponding diagonal matrix of eigenvalues where the
eigenvalues have been arranged in decreasing order, then selected the required number of
common factors from the matrix of eigenvectors. This procedure was programmed in MATLAB
to allow for a recursive out of sample forecasting exercise.
4.3 The Determination of Number of Factors using the Bai-Ng Test
We take two approaches to determine the number of factors to be used in our forecasting
models. The first approach is to test the performance of forecasting models with a fixed number
of variables, so as not to impose any prior methodology of determining the optimal factors to
include in the forecasting model. The second approach is the implementation of a selection
criterion to determine the optimal number of lags we want to include in our forecasting models.
There have been not many tests proposed to determine the number of common factors
driving the factor model. One popular criterion is the Bai-Ng (2002) test, which Stock and
Watson (2010) conducted limited Monte Carlo experiments upon and concluded that it had better
finite sample performance when compared with another popular alternative, the Amenguel-
Watson (2007) procedure. In this paper, the method we utilize is the Bai-Ng (2002) test to isolate
the number of common factors in the dataset.
In Stock and Watson (1998), the author used the Bayesian Information Criterion (BIC) to
determine the number of factors to include in their forecasting model. But Bai & Ng (2002)
showed that the BIC and Akaike’s Information Criterion (AIC) would overestimate the number
of factors required for factor analysis because the penalty function for over fitting was only a
function the number of observed periods T, and not of both T and the number of variables N.
Bai & Ng (2002) wanted an information criterion (IC) that consisted of two parts like (9).
The first term is such that the average residual variance decreases with increasing number of
factors, while the second term is a penalty function imposed on the IC that increases
with the number of factors due to over identification.
( ) (9)
Bai & Ng (2002) then proposed three different Information Criterions (IC) to determine
the optimal number of static factors:
( ( )) (
) (
)
(10)
( ( )) (
)
(11)
( ( ))
(12)
where {√ √ }, and ( )
∑ ∑
from equation (6)
Even though the three criteria are asymptotically equivalent, they have different
properties in finite samples; in fact, only (10) and (11) apply specifically to principal
components. Because Bai & Ng (2002, 2007) found that was effective as a test itself, we
utilized only when utilizing the Bai & Ng test in determining the number of factors. We
estimate the number of static factors in the model by choosing k such that , which
Bai & Ng (2002) demonstrated consistently estimates r, the true number of static factors.
5. Alternative Forecasting Models
The objective of this paper is to test the accuracy of factor model forecasts for the annual
inflation and IPI growth rate 1, 2, 3, 4 and 8 periods ahead. For notational purposes, we denote
the h-step ahead annualized inflation rate (where h = 1, 2, 3, 4, and 8) as:
[ (
)]
(13)
[ (
)]
.
(14)
where is the CPI at time t in the case of inflation forecasting, and is the Industrial
Production Index at time t in the case of forecasting IPI growth rates. is the h-step ahead
forecast of the target growth rate.
5.1 Atkeson-Ohanian (AO) Random Walk Model (Both Inflation and IPI Forecasting)
The AO random walk model is a univariate model that predicts that the forecast of the annual
rate of inflation or IPI growth rate is simply the average growth rate of the target variable over
the previous four quarters. Though Atkeson and Ohanian (2001) only considered the 4 period
ahead forecasts their random walk model, we extend this model to forecast 1, 2, 3, 4 and 8
quarters ahead. For example, if the average inflation for the end of the four quarters for the year
of 2002 was 3%, the forecasts for the next 1, 2, 3, 4 and 8 quarters ahead forecast would be 3%.
5.2 Direct Autoregressive (AR) Model (Both Inflation and IPI Forecasting)
The direct autoregressive forecast is obtained by ordinary least squares from the model:
∑
(15)
where h denotes the number of periods ahead we are forecasting, is a constant, denotes
the quarterly inflation rate, and is the h step ahead error term. Two methods were used to
determine the number of lags in the AR model: In the first model, the lag length at every forecast
period in the AR model was determined by the Bayesian Information Criterion with a maximum
of 6 lags. In a second set of models, a fixed number of one to six lags were used to calculate the
forecasts.
5.3 Multivariate Leading Indicator Model (Only Inflation Forecasting)
The multivariate leading indicator model is stated as:
∑
∑
(16)
where the parameters are estimated using ordinary least squares. h denotes the number of periods
ahead we are forecasting, denotes the quarterly inflation rate, }is a vector of the leading
indicators, is the h step ahead error term, and number of lags q and p were selected
separately by BIC (maximum lag of 6). The leading indicators were chosen following Stock and
Watson (1999) where the authors chose indicators that were proven to be effective for
forecasting purposes. We then selected leading indicators for Singapore based on Stock and
Watson (1999) as closely as possible where data was available. The leading indicators used are
the unemployment rate, forecasts of total new orders received, value of construction contracts
awarded, M1 money supply, overnight interbank rate, spread on overnight rate and T-bill, and
lastly the US dollar exchange rate.
5.4 Phillips Curve Model (Only Inflation Forecasting)
The Phillips Curve was first proposed by the New Zealand economist William Phillips, a
model that proposes an inverse and stable relationship between the level of unemployment and
the inflation level. The Phillips Curve gained popularity due to the policy implications the model
presented: If the Phillips Curve were true, then the government could target a certain level of
unemployment as means of ensuring price stability. Samuelson and Solow (1960) presented the
idea of a trade-off between unemployment and inflation, a trade-off policy makers could
capitalize on to achieve their macroeconomic goals. The original Phillips Curve model has been
modified over time to better explain observations of inflation and to improve inflation forecasts.
Criticism of the inability of the Phillips Curve to make reliable forecasts spurred economists
to make constant refinements to their models to try to account for policy and structural changes.
The Expectations-Augmented Phillips Curve and the Triangle model proposed by Gordon (1991)
are examples of such modifications of the original Phillips Curve. We use the second prototype
Phillips Curve model utilized in Stock and Watson (2008) which is an Autoregressive
Distributed Lag (ADL) model. The model is represented as:
∑
∑
(17)
where the parameters are estimated using ordinary least squares. h denotes the number of periods
ahead we are forecasting, denotes the quarterly inflation rate, the unemployment rate,
is the h step ahead error term, and number of lags q and p were selected separately by BIC
(maximum lag of 6).
5.5 Factor Model (Both Inflation and IPI Forecasting)
This model follows closely the diffusion index forecasting model used in Stock and Watson
(1998) of the form:
∑
∑
(18)
where { } are the estimated factors, is the quarterly growth rate at the period t-j, and is
the error term. The estimates for the coefficients of the diffusion index forecasting model were
calculated using OLS. We depart from Stock and Watson (1998) in using the Bai-Ng (2002) test
to select the optimal number of factors in the factor model instead of using BIC. We consider
four versions of (2):
i) The number of factors q in the forecasting model chosen at each period by the Bai-Ng
test, where , and no autoregressive components (p=0)
ii) The number of factors q recursively chosen by the Bai-Ng test, where ,
and the number of autoregressive components recursively chosen by BIC ( )
iii) No autoregressive components (p=0) but a fixed number of factors ( )
iv) A fixed number of factors( ) and autoregressive component p selected by
BIC ( )
6. Forecast Methodology
6.1.1 Forecast Comparison using the Giacomini-White (2006) Test
In order to evaluate whether the observed differences between the RMSEs calculated
from the forecasts are statistically significant, we implement a forecast evaluation to assess if a
particular forecast model statistically outperforms a benchmark model. Numerous tests have
been proposed by Diebold and Mariano (1995), West (1996), Clark and McCraken (2004), and
Giacomini and White (2006) to compare out of sample predictive ability for different forecasting
conditions. Popular tests such as the Diebold and Mariano (1995) and the West (1996) tests are
not directly applicable because these tests are applicable only in the comparison of non-nested
models. To best evaluate the statistical significance of the forecast performances of two
competing models, use the Giacomini-White (2006) test because it is a general test that permits a
comparison of both nested and non-nested models.
The test checks if there is a statistically significant difference in the forecasting
performance between two models. Thus the null hypothesis for the Giacomini-White equal
conditional predictive ability test is written as:
[ ( ) ( )| ] [ ] (19)
where is a loss function which in this case is the squared error, is the forecast horizon,
and are the two forecast model functions.
against the alternative:
for all sufficiently large n (20)
where ∑ , and is the test function. The alternative hypothesis can be
generally interpreted as the case where the differences in forecasting performance are not
statistically significant. In this paper, we use the same test function, , that
Giancomini & White (2006s) used in their tests to determine if the predicted diffusion index
forecasts that Stock and Watson (2002) generated were statistically significant. The flexibility1 in
selecting the test function allows the researcher to input his beliefs on how past relative
performance may help distinguish between the forecast performance in the future. In using
, we impose the beliefs that the difference in predictive ability today has potential
explanatory power for the future difference.
Lastly, the test statistic to be constructed is:
∑
∑
(21)
where n is the number of observations in the forecast period, ∑
∑ ∑
with a weight function such as
the one in Newey and West (1987). is also a consistent estimator of the variance of .
Lastly, the test statistic follows a distribution, which allows us to calculate the p-values of the
tests for statistical difference in forecasting performances.
6.1.2 Forecast Comparison using the Diebold-Mariano (1995) Test
Though the Diebold-Mariano (1995) test only allows us to make comparisons between non-
nested models, many empirical studies still implement the Diebold-Mariano (1995) test. It is
arguable that the AO Random-Walk model when compared to the other competing models is not
nested, thus the results of statistical significance obtained from this test should be accepted with
caution. Furthermore, since our forecast period is relatively short, we use Harvey, Leybourne and
Newbold (1997)’s small sample modification of the Diebold-Mariano (1995) test to evaluate
statistical significance.
1 In a paper where factor models were used to forecast inflation in Mexico (Ibarra-Ramirez 2010),
the author used a constant as an instrument when implementing the Giancomini & White (2003)
test.
The null hypothesis for the Diebold-Mariano (1995) test can be described as the equality
between the squared errors obtained from the two forecasting models being compared, and can
be stated as:
[ ] (22)
where is the same loss difference in the previous section.
We first define
∑
(23)
[ ∑
] (24)
∑
The modified Diebold-Mariano (1995) test statistic can then be written as:
(25)
[
]
[ ]
[
]
(26)
where n is the number of observations, is the kth
autocovariance of , and is the number
of periods ahead which we are forecasting. This test statistic can be compared against the critical
values from the Student’s t-distribution with degrees of freedom to allow us to calculate
the p-values of the tests for statistical difference in forecasting performances.
6.2 Estimation
In order to obtain the estimates of the forecasts from the model, we use the ‘direct’ forecast
method in which we regress a multiperiod-ahead value of the dependent variable on available
data from the past till today. Optimally, we would want to perform real-time forecasts, but such
forecasts are prohibitively complicated because of the large dataset involved which are often
announced with different lags and are subject to revision subsequently. However, we attempt to
closely replicate a real-time forecasting exercise by performing out-of-sample direct forecasts.
In the case of factor model forecasting, in order to obtain the forecast of period T+h where h
is the number of periods ahead we want to forecast, the data from period 1 to T was normalized
to avoid overweighing any variable in the dataset. After normalizing the data, the factors were
estimated by the method of principle components, the optimal number of factors and lags
selected by the Bai-Ng test and BIC respectively, and the regression coefficients were then
estimated from time 1 to T using OLS. The forecast was then constructed using the estimated
parameters and factors, and finally compared to the actual values for inflation at the forecast
horizon using an appropriate metric. To forecast the next period T+1+h, we repeat the procedure
using data from period 1 to T+1, reevaluating the procedure of calculating the information
criterions, factors, and estimators before making the forecast. The Diebold-Mariano (1995) test is
used in the fixed window estimation to evaluate the statistical significances of forecasting
performances.
The implementation of the Giacomini-White (2006) test described in section 6.1 required
the forecasting model to depend on forecasts generated on rolling window scheme. Instead of
adding a new observation to our dataset as in our first estimation exercise, we fix the estimation
window to 32 quarters in the rolling window exercise. For example, if we utilized data from
period 1 to period T to estimate forecasts at time T+h, we move the estimation window 1 quarter
ahead to data from time 2 to time T+1 to estimate forecasts at time T+h+1. The procedure to
calculating the information criterions, factors and estimators is similar to the previous exercise,
only the data used in the forecasts varies.
7. Results
7.1 Analyzing the Common Factors Extracted
Table 1 shows the values of the eigenvector and variance associated with factors 1
through 12. On the third column it also describes the cumulative percentage of variance
explained. From table 1 we can see that with the first 6 factors extracted from the
macroeconomic variables, we are able to explain almost half of the variation of our data set.
Moreover, the amount of variance explained by each factor is heavily skewed towards the first
three factors, explaining a cumulative 33.6% of total variance observed2. This result suggests that
there are only a few variables that are the source of most of the macroeconomic variability found
in Singapore, which leads us to try to identify the main sources of these variations.
Table 1. Variance explained by the first r common factors
r Eigenvector
value
Variance explained
by factor (%)
Cumulative percentage of
variance explained (%)
1 28.541 17.727 17.727
2 15.036 9.339 27.066
3 10.522 6.536 33.602
4 8.066 5.010 38.612
5 7.353 4.567 43.179
6 6.677 4.147 47.327
7 5.455 3.388 50.715
2 Similarly, a formal method utilizing a scree plot plots the eigenvectors for each factor in descending order, and the
optimal number of factors to be chosen is determined by the factor at the kink.
8 5.104 3.170 53.885
9 4.707 2.924 56.808
10 4.216 2.618 59.427
11 3.919 2.434 61.860
12 3.568 2.216 64.077
The factors extracted from the factor model itself do not have explicit interpretations, and
moreover could just be linear combinations of the variables from the time series. But because
these factors span the same space as the structural factors (Kotlowski 2008), we adopt Stock and
Watson’s (2002b) methodology of regressing each of the individual time series against each of
the extracted common factors and finding the coefficient of determination, or , for each
regression. Stock and Watson (2002b) used the values of the associated with each variable to
evaluate the composition of each factor; the higher the values of , the better the particular
factor explains a specific variable. In Figure 3, the s of the individual variables corresponding
to the top 5 common factors are plotted. On the horizontal axis are the 161 variables, and on the
vertical axis is the value of the R2s. The vertical lines that partition the bars separate the variables
separate into the general sectors that were previously defined in section 3.
Figure 3. From the Regressions of the Variables on the Top 5 Factors
From figure 3, we see that Factor 1 is characterized mainly by three groups of
macroeconomic time series: The bulk of the factor is driven by trade related variables such as
import and export of goods and services, as well as economic indicators such as the Composite
Leading Index and GDP of Singapore’s major trading partners. Lastly, the factor is also driven
by business expectations of the Singapore economy, for example the variables on expectations
on the manufacturing sector and the total new orders received. The characterization of the main
factor driving the variability reflects Singapore’s economy that is historically and currently
driven by exports and manufacturing because of her strategic location and lack of natural
resources.
Factor 2 is driven by a different set of variables, most notably by variables that are
closely linked to the labor market, as well as the business expectations on the performance of
Singapore’s major industries. Lastly, the second factor loads the variables related to the number
of companies formed in Singapore. The 3rd
and 4th
factors contribute less to the total variance
explained, and can be broadly described as factors pertaining to Singapore’s financial indicators
and company formation respectively. Again, we take caution in interpreting the common factors,
but further research might allow for some insight into not only what indicators really drive these
common factors, as well as policy implications, if any, that can be derived from the interpretation
of these common factors.
7.2 Empirical Results
The results of the forecasting experiment are listed in Table 1-5, where we evaluate the
performance of each forecasting model by the Root Mean Squared Error (RMSE) defined as:
∑
(22)
where
are the observed values of the h period ahead growth rates,
are the
estimated forecasts of growth rates, and T is the number of periods being forecasted in the
exercise. Appendix B and C shows the RMSE for the AO random walk model and relative
performances of the other forecasting models. The entries in the appendix are the RMSE of the
candidate forecasting model as a ratio of the RMSE of the benchmark forecasting model. A ratio
of less than 1 signifies more accurate forecasts, while a ratio of more than 1 would mean that the
model in question did not outperform a simple random walk. Appendix B shows the results from
a fixed window inflation forecasting exercise, and Appendix C shows the results of utilizing a
rolling window in the inflation forecasting exercise. Appendix D and E shows the results for IPI
growth forecasting for the fixed and rolling window estimates respectively.
7.2.1 Fixed vs Rolling Window Forecasts
Economic forecasting literature have recognized the trade-offs between using a fixed or
rolling window in data selection. Clark and McCracken (2004) explained that a rolling window
forecast would allow researchers to avoid the biases associated with structural change, but the
smaller amount of observable data would mean a larger variance in parameter estimates.
For inflation forecasts using both a fixed window as well as a rolling window, we find
that for factor forecasting models, the rolling window estimates generally outperformed the fixed
window estimates at the 3 and 4 quarter ahead forecasts. For the rest of the models, the fixed
window inflation forecast estimates generally outperformed the rolling window estimates at all
forecasting horizons.
For the case of IPI growth, the results were slightly different. For the AR models, fixed
window forecasts outperformed rolling window forecasts. For the factor models with lags, the
fixed window forecasts outperformed almost all of the rolling window forecasts. For factor
models without lags, the rolling window forecasts performed better in the long range horizons of
3, 4, and 8 step ahead forecasts. In summary, it is difficult to pinpoint an exact relationship
between the forecast performance using either estimation windows for all the models, which
reflects the ambiguity and trade-offs from using both methods.
7.2.2 Evaluation of the Forecasting Models
Firstly, we outline the general results we find from both rolling and fixed window
forecasting exercises. At the 1 period ahead horizon, the AO random walk model outperformed
all our forecasting models. For the inflation forecasts, at the 3, 4 and 8 quarters ahead horizons,
our factor forecasting models generally outperformed the AO random walk for both the fixed and
rolling window procedures. For the IPI growth forecasts, the only improvement was seen at the 8
quarter ahead horizon, the improvement at this horizon was more than the improvement for
inflation forecasting. A more significant finding would be that the inclusion of lags in the factor
models actually caused the performance of the factor model to deteriorate significantly compared
to the factor-only model without any lags.
7.2.2.1 Inflation Forecasting
We found mixed results in performance of factor models for inflation forecasting and IPI
growth forecasting. In the case of inflation forecasting, we see that under fixed window
estimation, the factor model without lags that used one factor produced the best overall
performance. This result is surprising because the first factor only accounts for 17.27% of total
variance explained in the dataset, and we would expect using more factors to improve forecast
performance. Under the rolling window procedure, the factor-only model that used the Bai-Ng
test to select the optimal number of factors showed the most accurate forecasts. This result is the
same as Stock and Watson (1999, 2002) that factor models do indeed improve inflation forecast
accuracy.
As for the competing models, we find that the AR model also outperformed the AO
random walk model at the 4 and 8 quarter ahead horizons. However, for the MLI model and
Phillips Curve model, it seems that the ability to consistently make accurate forecasts broke
down. They outperform the AO random walk model at the 3 and 4 quarter ahead forecast when
using the fixed window estimation, and outperform the AO random walk model only at the 8
quarter ahead forecast when using the rolling window estimation. It is interesting to analyze if
the Phillips Curve and MLI model still has predictive abilities in the Singaporean context, and
warrants further research.
7.2.2.2 IPI Growth Forecasting
For IPI growth forecasting, the factor-only models produced the best forecasts amongst
the factor augmented forecasting models, but surprisingly did not show a significant
improvement from the AR model. As mentioned in a previous section, the inclusion of lags
caused forecasting performance to deteriorate, and in the case of IPI growth forecasting, the
factor models with lags performed worse than the AR model.
7.2.3 Selection of the Optimal Number of Factors and Lags
In recent literature with regards to macroeconomic forecasting, many papers utilized
information criterions such as the BIC, AIC and the Bai-Ng test to determine the optimal number
of lags and factors in their dynamic factor forecasting models. In this paper, we ran forecasting
models that utilized these criterions, as well as ran a range of models of all the possible
combinations of lags and factors. Thus it would be insightful to see if these criterions actually
improve forecasts for Singapore, for both factor models as well as the other models utilized in
this paper.
We find that for the inflation forecasting model with factors and lags of inflation using
the fixed window procedure, using the Bai-Ng test to select the optimal number of factors and
then the BIC to select the optimal number of lags resulted in a significant improvement in
forecasting performance especially in the 1, 2, 4 and 8 quarter ahead horizons. Under the rolling
window procedure, the utilization of the Bai-Ng test in the factors-only model improved upon
the inflation forecasting performances of the fixed-factors-only model.
However, in the rest of the models that used the Bai-Ng test to forecast inflation or IPI
growth, the implementation of the Bai-Ng test did not result in any conclusive improvement in
forecasting ability. We see that the implementation of the Bai-Ng test gave mixed results, and
further research into different selection techniques should be performed to see which selection
model would improve upon the fixed-factor forecast model.
7.2.4 Statistical Significance of Improvements in Forecasting Ability
The numbers in parenthesis below the ratio of RMSE are the p-value from the Diebold-
Mariano or Giancomini-White test statistic. We see that when the inflation and IPI growth
forecast models are compared to the AO Random Walk model, in the cases where we see an
improvement in forecast performance, it is generally significant only at the 8 quarter ahead
horizon where there is great improvement in forecasting performance. This is the case for the AR
models, Phillips curve model, and some of the variations of the models that include common
factors in the forecast model.
In section 7.2.2.2, we showed that there was no obvious improvement in IPI growth
forecasting performance when we included factors in the forecasting models. We switch our
focus towards inflation forecasting models, and to better evaluate their effectiveness, we
compare the factor model forecasts not only to the AO random walk model, but to the other
models which have shown the best performances.
Table 2. Comparisons of 4 Quarter Ahead Forecasts for Inflation
AO
Random
Walk
MLI
Forecast
Phillips
Curve
AR
Model,
Lags
chosen by
BIC
Factor
Selection
using Bai-
Ng
Lag
Selection
using BIC
and Factor
Selection
using Bai-
Ng
AO Random - 0.606023 1.172285 1.208294 1.381087 1.303677
Walk (0.176549) (0.542642) (0.521506) (0.426565) (0.475526)
MLI Forecast
1.650102
(0.176549) -
1.93439
(0.062892)
1.993808
(0.062425)
2.278935
(0.068546)
2.151201
(0.069162)
Phillips Curve
0.853035
(0.542642)
0.516959
(0.062892) -
1.030717
(0.095048)
1.178116
(0.165976)
1.112083
(0.630646)
AR Model,
Lags chosen by
BIC
0.827613
(0.521506)
0.501553
(0.062425)
0.970198
(0.095048) -
1.143006
(0.333762)
1.078941
(0.761078)
Factor Selection
using Bai-Ng
0.724067
(0.426565)
0.438801
(0.068546)
0.848813
(0.165976)
0.874886
(0.333762) -
0.94395
(0.212215)
Lag Selection
using BIC and
Factor Selection
using Bai-Ng
0.767061
(0.475526)
0.464857
(0.069162)
0.899214
(0.630646)
0.926835
(0.761078)
1.059378
(0.212215) -
Table 2 is a tabular comparison of two versions of the factor forecasting models as well
as the AO random walk model, MLI model, Phillips Curve model, and the AR model. We only
make comparisons of the 4 quarter ahead forecast, and we only consider the results in the lower
diagonal of the symmetric 6x6 table. The numbers in the first row of each cell shows the ratio of
the Root Mean Squared Errors (RMSE of the row model/RMSE of the column model. From the
table, we see that the factor models do indeed outperform all the other models, but the
improvements from using the factor models are not statistically significant at the 10%
significance level, with the exception of the MLI model. Thus we see that even though factor
forecasting models do outperform their peers under the RMSE comparison metric, we should
take caution in accepting these results from the outset because the improvements in their
forecasting ability is not statistically significant.
8. Conclusions
In this study, we examine the out-of-sample forecasting performance of factor models in the
case of Singapore for the period from 1992:Q2 to 2009 Q4. We described the variables gathered,
the models used, the methodology to achieve the forecasts, and the forecast evaluation
techniques, to ultimately determine if the inclusion of factor analysis could improve upon
conventional forecasting models. Comparing the results of the factor forecasting model, we
reached three main conclusions.
Firstly, all the factor models we used in the exercise outperformed the AO Random Walk
model only at the longer forecast horizons. Only in the case of inflation forecasting, did factor
models outperform all the other models, signifying increased accuracy. In the case of IPI growth
forecasting, factor models did not contribute to any improvement in forecasting ability. Secondly,
the inclusion of the Bai-Ng test to select the optimal number of factors did not conclusively
improve forecast performance in all our forecasting exercises. And lastly, even though the use of
factor models did indeed result in lower RMSE in forecasting performance when compared to
the random walk model, the difference is only significant at the 8 quarters ahead horizon. Thus
there is a possibility that the improvement in performance when using factor models could be
attributed to chance, and this problem can be resolved with a greater number of observations.
We conclude that there is some evidence that factor models are useful for inflation but not for
IPI growth forecasting purposes in the case of Singapore. More analysis should be done to
determine why the performances of factor driven inflation forecasting models differ across the
variables to be forecasted. One hypothesis could be that the factor model does not perform as
well when the variable to be forecasted is extremely volatile. Further improvements could be
made to the factor models by involving factor analysis in non-parametric models, the use of a
larger more comprehensive data set, or a more effective factor selection technique, which might
improve the performance of factor models and become a reliable tool for policymakers around
the world.
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Appendix A
Description Mnemonic Tran
1) Real GDP
1 GROSS DOMESTIC PRODUCT AT CURRENT
MARKET PRICES
GDPMKTPRICE ∆ln
2 EXPENDITURE ON GDP EXPGDP ∆ln
3 PRIVATE CONSUMPTION EXPENDITURE PVTCONEXP ∆ln
4 GOVERNMENT CONSUMPTION EXPENDITURE GOVCONEXP ∆ln
5 GROSS FIXED CAPITAL FORMATION FXCAPFORM ∆ln
6 GROSS FIXED CAPITAL FORMATION
CONSTRUCTION & WORKS
FXCAPFORMCONSTRUCT ∆ln
7 GROSS FIXED CAPITAL FORMATION
RESIDENTIAL BUILDINGS
FXCAPFORMRESBUILD ∆ln
8 GROSS FIXED CAPITAL FORMATION NON-
RESIDENTIAL BUILDINGS
FXCAPFORMNONRESBUILD ∆ln
9 GROSS FIXED CAPITAL FORMATION OTHER
CONSTRUCTION
FXCAPFORMOTHERCONSTRUCT ∆ln
2) Trade Indicators
10 EXPORTS OF GOODS AND SERVICES EXGDAS ∆ln
11 IMPORTS OF GOODS AND SERVICES IMGDAS ∆ln
12 EXTERNAL TRADE: TOTAL EXTRADETOTAL ∆ln
13 EXTERNAL TRADE: TOTAL OIL EXTRADETOTALOIL ∆ln
14 EXTERNAL TRADE: TOTAL IMPORTS EXTRADETOTALM ∆ln
15 EXTERNAL TRADE: TOTAL IMPORTS OIL EXTRADEOILM ∆ln
16 EXTERNAL TRADE: IMPORTS NON OIL EXTRADENONOILM ∆ln
17 EXTERNAL TRADE: TOTAL EXPORTS EXTRADETOTALX ∆ln
18 EXTERNAL TRADE: OIL EXPORT EXTRADEOILX ∆ln
19 EXTERNAL TRADE: NON OIL EXPORT EXTRADENONOILX ∆ln
20 EXTERNAL TRADE: TOTAL REEXPORTS EXTRADERE ∆ln
21 EXTERNAL TRADE: REXPORT OIL EXTRADEREOIL ∆ln
22 EXTERNAL TRADE: REXPORT NON OIL EXTRADERENONOIL ∆ln
3) Foreign Exchange Rate
23 US DOLLAR USD ∆ln
24 STERLING POUND STER ∆ln
25 JAPANESE YEN JPY ∆ln
26 MALAYSIAN RINGGIT MALAYRM ∆ln
27 HONG KONG DOLLAR HKD ∆ln
28 KOREAN WON WON ∆ln
29 NEW TAIWAN DOLLAR NTD ∆ln
30 INDONESIAN RUPIAH INDO ∆ln
31 THAI BAHT BAHT ∆ln
4) Price Indices
32 GROSS DOMESTIC PRODUCT DEFLATORS AT
MARKET PRICE
GDPDEFFLATORMKTPX ∆2 ln
33 GROSS DOMESTIC PRODUCT DEFLATORS
MANUFACTURING
GDPDEFFLATORMANU ∆2 ln
34 GROSS DOMESTIC PRODUCT DEFLATORS
CONSTRUCTION
GDPDEFFLATORCONSTRUCT ∆2 ln
35 GROSS DOMESTIC PRODUCT DEFLATORS
SERVICE
GDPDEFFLATORSERVICE ∆2 ln
36 DOMESTIC SUPPLY PRICE INDEX DOMSUPPI ∆2 ln
37 IMPORT PRICE INDEX: OVERALL MPIOVERALL ∆2 ln
38 IMPORT PRICE INDEX: NONOIL MPINONOIL ∆2 ln
39 EXPORT PRICE INDEX: OVERALL XPIOVERALL ∆2 ln
40 EXPORT PRICE INDEX: NONOIL XPINONOIL ∆2 ln
41 CPI: ALL ITEMS CPIALL ∆2 ln
42 CPI: FOOD CPIFOOD ∆2 ln
43 CPI: CLOTHING CPICLOTHES ∆2 ln
44 CPI: TRANSPORT CPITRAN ∆2 ln
45 CPI: HEALTH CPIHEALTH ∆2 ln
46 CPI: HOUSEHOLD ITEMS CPIHHG ∆2 ln
47 CPI: NONDURABLE HOUSEHOLD GOODS CPINDHHG ∆2 ln
5) Sectorial Indicators
48 FORMATION OF COMPANIES: REAL ESTATE,
RENTAL AND LEASING ACTIVITIES
FORMRE ∆ln
49 FORMATION OF COMPANIES: TOTAL FORMTOTAL ∆ln
50 FORMATION OF COMPANIES:
MANUFACTURING
FORMMAN ∆ln
51 FORMATION OF COMPANIES: CONSTRUCTION FORMCONSTR ∆ln
52 FORMATION OF COMPANIES: WHOLESALE
AND RETAIL COMPANIES
FORMWRC ∆ln
53 FORMATION OF COMPANIES: TRANSPORT
AND STORAGE
FORMTRANS ∆ln
54 FORMATION OF COMPANIES: HOTELS AND
RESTAURANTS
FORMHOTEL ∆ln
55 FORMATION OF COMPANIES: INFO AND
COMM
FORMINFO ∆ln
56 FORMATION OF COMPANIES: FINANCIAL AND
INSURANCE
FORMFIN ∆ln
57 VISTOR ARRIVAL VISTOR ∆ln
58 RETAIL SALES INDEX RETAILSI ∆ln
59 AIR CARGO DISCHARGED AIRCARGO ∆ln
60 NEW REGISTRATION OF MOTOR VEHICLES MOTOR ∆ln
61 SEA PASSENGERS SEAPASSENGERS ∆ln
62 COMPOSITE LEADING INDEX COMLI ∆ln
63 ELECTRICITY GENERATION AND SALES ELECTRICITY ∆ln
6) Labor Market
64 RETRENCHED WORKERS RETRENCH ∆ln
65 UNIT LABOUR COST INDEX LABCOSTS ∆ln
66 UNIT BUSINESS COST INDEX OF
MANUFACTURING: OVERALL
BIZCOSTOVERALL ∆ln
67 UNIT BUSINESS COST INDEX OF
MANUFACTURING: MANUFACTURING
BIZCOSTMANU ∆ln
68 EMPLOYMENT BY SECTOR: TOTAL EMPLOYALL ∆ln
69 UNEMPLOYMENT RATE: TOTAL UNEMPRATE ∆level
70 RESIDENT UNEMPLOYMENT RATE UNEMPRES ∆level
71 AVERAGE MONTHLY EARNINGS: TOTAL WAGETOTAL ∆2 ln
72 EMPLOYMENT CHANGES: SERVICES SECTOR
TOTAL EMPLOYMENT
EMPCHANGETOTAL Level
73 EMPLOYMENT CHANGES: SERVICES SECTOR
WHOLESALE AND RETAIL TRADE
EMPLOYMENT
EMPCHANGEWRT Level
74 EMPLOYMENT CHANGES: HOTELS AND
CATERING EMPLOYMENT
EMPCHANGEHOTEL Level
75 EMPLOYMENT CHANGES: TRANSPORT AND
STORAGE EMPLOYMENT
EMPCHANGETRANS Level
76 EMPLOYMENT CHANGES: FINANCIAL
SERVICES EMPLOYMENT
EMPCHANGEFIN Level
77 EMPLOYMENT CHANGES: REAL ESTATE
EMPLOYMENT
EMPCHANGERE Level
78 EMPLOYMENT CHANGES: BUSINESS EMPCHAGNEBIZ Level
SERVICES EMPLOYMENT
7) Construction
79 HDB RESALE PRICE INDEX HDBPX ∆ln
80 PROPERTY PRICE INDEX: RESIDENTIAL PROPPIRES ∆ln
81 PROPERTY PRICE INDEX: OFFICE PROPPIOFFICE ∆ln
82 PROPERTY PRICE INDEX: SHOP SPACE PROPPISHOP ∆ln
83 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: TOTAL PUBLIC
& PRIVATE SECTOR
CONTRACTTOTAL ∆ln
84 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: PUBLIC
SECTOR
CONTRACTPUBLIC ∆ln
85 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: PRIVATE
SECTOR
CONTRACTPRIVATE ∆ln
86 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: RESIDENTIAL
BUILDING
CONTRACTRES ∆ln
87 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: COMMERCIAL
BUILDING
CONTRACTCOMM ∆ln
88 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: INDUSTRIAL
BUILDING
CONTRACTINDUS ∆ln
89 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK: CIVIL
ENGINEERING
CONTRACTCIVIL ∆ln
90 VALUE OF CONTRACTS AWARDED BY
SECTOR AND TYPE OF WORK:
INSTITUTIONAL
CONTRACTINSTITUTE ∆ln
8) Industrial Production
91 INDEX OF INDUSTRIAL PRODUCTION: TOTAL IIPTOTAL ∆ln
92 INDEX OF INDUSTRIAL PRODUCTION:
PETROLEUM
IIPPETRO ∆ln
93 INDEX OF INDUSTRIAL PRODUCTION:
CHEMICAL
IIPCHEM ∆ln
94 INDEX OF INDUSTRIAL PRODUCTION:
PHARMACEUTICALS
IIPPHARM ∆ln
95 INDEX OF INDUSTRIAL PRODUCTION:
MACHINERY
IIPMACHINE ∆ln
96 INDEX OF INDUSTRIAL PRODUCTION:
ELECTRICAL MACHINERY
IIPELECMACH ∆ln
97 INDEX OF INDUSTRIAL PRODUCTION:
ELECTRONICS
IIPELECTRONIC ∆ln
98 INDEX OF INDUSTRIAL PRODUCTION:
TRANSPORT
IIPTRANS ∆ln
99 INDEX OF INDUSTRIAL PRODUCTION:
FURNITURE & MANUFACTURING
IIPFURN ∆ln
9) Monetary Indicators
100 MONEY SUPPY: M3 MTHREE ∆2 ln
101 MONEY SUPPY: M2 MTWO ∆2 ln
102 MONEY SUPPY: M1 MONE ∆2 ln
103 MONEY SUPPY: CURRENCY CURRENCY ∆2 ln
104 MONEY SUPPY: DEMAND DEPOSITS DEMANDD ∆2 ln
105 MONEY SUPPY: FIXED DEPOSITS FIXEDD ∆2 ln
106 BANK LOANS: TOTAL BANKTOTAL ∆2 ln
107 BANK LOANS: MANUFACTURING BANKMANU ∆2 ln
108 BANK LOANS: BUILDING BANKBUILD ∆2 ln
109 BANK LOANS: GENERAL COMMERCE BANKCOMM ∆2 ln
110 BANK LOANS: FINANCIAL INSTITUTIONS BANKFIN ∆2 ln
111 BANK LOANS: PROFESSIONAL AND PRIVATE
INDIVIDUALS
BANKINDIV ∆2 ln
10) Indicators
112 PRIME LENDING RATE PRIMELENDING Level
113 3-MONTH INTERBANK RATE THREEIBR Level
114 3-MONTH US$ SIBOR THREEUS Level
115 GOVERNMENT SECURITIES - 3-MONTH
TREASURY BILLS YIELD
THREEMONTHTBILL Level
116 GOVERNMENT SECURITIES - 5-YEAR BOND
YIELD
FIVEYEARBOND Level
117 GOVERNMENT SECURITIES - 2-YEAR BOND
YIELD
TWOYEARBOND Level
118 3-MONTH COMMERCIAL BILLS THREEMONTHCBILL Level
119 GOVERNMENT SECURITIES - 1-YEAR
TREASURY BILLS YIELD
ONEYEARTBILL Level
11) Business Expectations:
120 BUSINESS EXPECTATIONS OF THE
MANUFACTURING SECTOR: GENERAL
BUSINESS EXPECTATIONS (FORECAST FOR
NEXT 6 MONTHS)
BIZEXPGEN Level
121 BIZ EXP: TOTAL NEW ORDERS RECEIVED
(FORECAST FOR NEXT QUARTER)
BIZNEWORDER Level
122 BIZ EXP: EXPORT ORDERS (FORECAST FOR
NEXT QUARTER)
BIZEXPEXPORT Level
123 BIZ EXP: DELIVERIES IN SINGAPORE
(FORECAST FOR NEXT QUARTER)
BIZEXPDEL Level
124 BIZ EXP: DELIVERIES OVERSEAS (FORECAST
FOR NEXT QUARTER)
BIZEXPDELO Level
125 BIZ EXP: WHOLESALE & RETAIL TRADE
ENDING STOCKS FORECAST
BIZEXPWRTENDSTOCKFOR Level
126 BIZ EXP: WHOLESALE & RETAIL TRADE
ENDING STOCKS PERFORMANCE
BIZEXPWRTENDSTOCKPER Level
127 BUSINESS EXPECTATIONS FOR THE SERVICES BIZEXPHOTELSENDSTOCK Level
SECTOR (PERFORMANCE) - ENDING STOCKS
OF MERCHANDISE, BIZ EXP: HOTELS &
CATERING
128 BUSINESS EXPECTATIONS FOR THE SERVICES
SECTOR (FORECAST FOR THE NEXT
QUARTER) - GENERAL BUSINESS
BIZEXPSERTOTALFOR Level
129 BIZ EXP: TOTAL SERVICES SECTOR
(PERFORMANCE)
BIZEXPTOTALPER Level
130 BIZ EXP: WHOLESALE & RETAIL TRADE
(PERFORMANCE)
BIZEXPWRTPER Level
131 BIZ EXP: HOTELS & CATERING
(PERFORMANCE)
BIZEXPHOTELPER Level
132 BIZ EXP: TRANSPORT & STORAGE
(PERFORMANCE)
BIZEXPTRANSPER Level
133 BIZ EXP: FINANCIAL SERVICES
(PERFORMANCE)
BIZEXPFINPER Level
134 BIZ EXP: BANKS & FINANCE COMPANIES
(PERFORMANCE)
BIZEXPBANKSPER Level
135 BIZ EXP: REAL ESTATE (PERFORMANCE) BIZEXPREPER Level
136 BIZ EXP: BUSINESS SERVICES
(PERFORMANCE)
BIZEXPBIZPER Level
12) FOREIGN COMPOSITE LEADING INDEX
137 COMPOSITE LEADING INDEX GERMANY COMPIGERMANY ∆ln
138 COMPOSITE LEADING INDEX JAPAN COMPIJAPAN ∆ln
139 COMPOSITE LEADING INDEX UNITED
KINGDOM
COMPIUK ∆ln
140 COMPOSITE LEADING INDEX UNITED STATES COMPIUS ∆ln
141 COMPOSITE LEADING INDEX EURO AREA COMPIEURO ∆ln
142 COMPOSITE LEADING INDEX FOUR BIG
EUROPEAN
COMPIBIGFOUREURO ∆ln
143 COMPOSITE LEADING INDEX KOREA COMPIKOREA ∆ln
144 COMPOSITE LEADING INDEX MAJOR FIVE
ASIA
COMPIASIA ∆ln
13) FOREIGN STOCK PRICES
145 STOCK PRICE: AUSTRALIA STOCKAUS ∆ln
146 STOCK PRICE: JAPAN STOCKJAP ∆ln
147 STOCK PRICE: EURO AREA STOCKEURO ∆ln
148 STOCK PRICE: KOREA STOCKKOREA ∆ln
149 STOCK PRICE: UNITED KINGDOM STOCKUK ∆ln
150 STOCK PRICE: UNITED STATES STOCKUS ∆ln
14) FOREIGN GDP
151 QUARTERLY GROWTH RATES OF GDP:
AUSTRALIA
AUSGDP Level
152 QUARTERLY GROWTH RATES OF GDP: JAPAN JAPGDP Level
153 QUARTERLY GROWTH RATES OF GDP: KOREA KOREAGDP Level
154 QUARTERLY GROWTH RATES OF GDP:
UNITED KINGDOM
UKGDP Level
155 QUARTERLY GROWTH RATES OF GDP: USA USAGDP Level
Appendix B Inflation Forecast Using Fixed Window
Table 1
AO Random-Walk Model Number of quarters ahead
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
RMSE 0.020156 0.025432 0.028614 0.029474 0.024249
Table 2
Direct Autoregressive Model Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1.437441 1.082763 0.861052 0.746242 0.662707
(0.04752) (0.34619) (0.21916) (0.06851) (0.06391)
2 1.511935 1.128133 0.869112 0.735881 0.664973
(0.03841) (0.26439) (0.21279) (0.07523) (0.06033)
3 1.573489 1.166454 0.863166 0.73078 0.665615
(0.0305) (0.19821) (0.20401) (0.07886) (0.05652)
4 1.604399 1.170219 0.852184 0.739614 0.66133
(0.02255) (0.182) (0.20205) (0.08309) (0.04943)
5 1.596307 1.170251 0.839326 0.736606 0.66841
(0.01945) (0.15423) (0.20017) (0.08749) (0.04617)
BIC 1.437441 1.082763 0.861052 0.746242 0.670956
(0.04752) (0.34619) (0.21916) (0.06851) (0.03491)
Table 3
Multivariate Leading Index
Model
Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC 1.466259 1.124545 0.902328 0.787422 1.198649
(0.04722) (0.27222) (0.31621) (0.09596) (0.3543)
Table 4
Autoregressive Phillips Curve
Model
Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC 1.462022 1.131981 0.91325 0.792831 1.018858
(0.04542) (0.26828) (0.33404) (0.09502) (0.47926)
Table 5
Factor Model Version (i) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
0 Bai-Ng
Test 1.353218 0.962655 0.815328 0.714488 0.632438
(0.08593) (0.44101) (0.21803) (0.0999) (0.0657)
Table 6
Factor Model Version (ii) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC Bai-Ng
Test 1.332313 1.047948 0.927169 0.75452 0.670007
(0.09485) (0.41943) (0.38542) (0.10174) (0.05395)
Table 7
Factor Model Version (iii) Number of periods ahead
No Lags Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1.364057 0.949791 0.784691 0.702258 0.628475
(0.07493) (0.42709) (0.20041) (0.09864) (0.06468)
2 1.351544 0.968871 0.816181 0.711611 0.630555
(0.08682) (0.45089) (0.21966) (0.09808) (0.06588)
3 1.355728 0.959205 0.817838 0.712501 0.62742
(0.08665) (0.43544) (0.21916) (0.09851) (0.06571)
4 1.38229 0.952813 0.844978 0.722386 0.63507
(0.0834) (0.42295) (0.24171) (0.09251) (0.06311)
5 1.371731 0.934652 0.849274 0.724605 0.64294
(0.08475) (0.38994) (0.23599) (0.08913) (0.06847)
6 1.38112 0.943741 0.842564 0.724655 0.641038
(0.0865) (0.40159) (0.22392) (0.0881) (0.06599)
7 1.387655 0.9426 0.852948 0.736552 0.636935
(0.07776) (0.39855) (0.23029) (0.08388) (0.07675)
8 1.393868 0.944583 0.845972 0.740791 0.656799
(0.07454) (0.40382) (0.23166) (0.08057) (0.0808)
9 1.391421 0.957341 0.851686 0.738103 0.644018
(0.07659) (0.42332) (0.24309) (0.08311) (0.09195)
10 1.377824 0.961075 0.852893 0.747824 0.653063
(0.07531) (0.43067) (0.24705) (0.08775) (0.09596)
11 1.351278 0.957908 0.86032 0.747148 0.663278
(0.10219) (0.42114) (0.24529) (0.08932) (0.08436)
12 1.343455 0.96491 0.854101 0.746076 0.655289
(0.11024) (0.43301) (0.23092) (0.08162) (0.07591)
Table 8
Factor Model Version (iv) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1 1.816758 1.321669 1.03525 0.881352 0.692457
(0.01255) (0.06708) (0.41109) (0.07732) (0.04498)
2 1 1.37487 1.071865 0.85128 0.72722 0.641931
(0.08773) (0.38627) (0.26485) (0.10238) (0.06193)
3 1 1.407145 1.097433 0.837657 0.734639 0.637413
(0.07392) (0.3497) (0.2483) (0.10361) (0.05221)
4 1 1.432188 1.115573 0.842646 0.749476 0.642086
(0.05531) (0.32442) (0.25687) (0.10503) (0.04976)
5 1 1.461427 1.130405 0.864411 0.77149 0.650059
(0.04131) (0.30608) (0.2807) (0.10612) (0.05139)
1 2 1.853861 1.32069 1.043773 0.89913 0.693695
(0.01066) (0.07655) (0.3901) (0.07428) (0.04649)
2 2 1.380643 1.113997 0.894158 0.75998 0.641027
(0.07929) (0.32912) (0.33056) (0.10196) (0.06138)
3 2 1.414435 1.149465 0.901097 0.794862 0.632801
(0.07189) (0.29527) (0.34134) (0.09251) (0.05244)
4 2 1.449473 1.176981 0.912222 0.826268 0.640971
(0.05405) (0.26829) (0.35632) (0.08959) (0.05024)
1 3 1.853777 1.31527 1.046354 0.901715 0.697158
(0.0115) (0.0794) (0.38177) (0.07469) (0.0456)
2 3 1.383693 1.100321 0.888783 0.76218 0.639413
(0.08135) (0.34746) (0.3236) (0.10302) (0.05677)
3 3 1.413853 1.147675 0.900301 0.790072 0.637616
(0.07087) (0.29836) (0.33945) (0.09319) (0.05669)
4 3 1.446208 1.167081 0.922494 0.84843 0.634896
(0.05492) (0.28039) (0.37214) (0.07466) (0.04849)
5 3 1.588829 1.25499 0.965724 0.905513 0.649239
(0.02086) (0.23298) (0.45297) (0.09027) (0.05072)
1 4 1.820791 1.338801 1.089253 0.930433 0.7156
(0.00783) (0.06563) (0.23273) (0.07226) (0.04129)
2 4 1.338157 1.114636 0.92399 0.795775 0.67177
(0.10537) (0.324) (0.3727) (0.08677) (0.06626)
3 4 1.370704 1.167633 0.944607 0.818331 0.641991
(0.09791) (0.27866) (0.40182) (0.08428) (0.05594)
4 4 1.416726 1.18356 0.940528 0.850933 0.639892
(0.07937) (0.24556) (0.39937) (0.07387) (0.04956)
5 4 1.608559 1.242392 0.983038 0.927239 0.650588
(0.01547) (0.234) (0.47679) (0.08873) (0.05452)
1 5 1.822867 1.329217 1.10606 0.929592 0.717666
(0.00792) (0.06395) (0.13993) (0.06922) (0.0475)
2 5 1.340349 1.085953 0.915468 0.785171 0.663785
(0.10241) (0.3619) (0.34809) (0.088) (0.06907)
3 5 1.37766 1.145702 0.936494 0.811703 0.650711
(0.09278) (0.30233) (0.37699) (0.08048) (0.0589)
4 5 1.42335 1.14657 0.908064 0.824059 0.635766
(0.07386) (0.286) (0.34148) (0.07274) (0.05166)
5 5 1.621096 1.264109 1.002998 0.883927 0.678098
(0.01525) (0.22188) (0.49586) (0.09179) (0.08837)
1 6 1.819798 1.333948 1.095749 0.929668 0.724938
(0.00709) (0.05599) (0.15838) (0.0689) (0.04906)
2 6 1.372485 1.090525 0.903819 0.777229 0.654584
(0.08866) (0.35761) (0.32873) (0.08805) (0.06776)
3 6 1.389433 1.164811 0.930669 0.806332 0.641915
(0.07467) (0.27218) (0.3594) (0.07989) (0.056)
4 6 1.431628 1.167623 0.91085 0.825281 0.637084
(0.06247) (0.25327) (0.34341) (0.07122) (0.05416)
5 6 1.62635 1.272419 0.989551 0.879988 0.682607
(0.01405) (0.21202) (0.4851) (0.0914) (0.08873)
1 7 1.843806 1.344365 1.103709 0.969191 0.735371
(0.00876) (0.05493) (0.18629) (0.06651) (0.03691)
2 7 1.356286 1.113877 0.909172 0.779272 0.649878
(0.09326) (0.32509) (0.33769) (0.08288) (0.07041)
3 7 1.422158 1.170617 0.919829 0.804949 0.650682
(0.04842) (0.26643) (0.33835) (0.07667) (0.05568)
4 7 1.441339 1.184437 0.91485 0.829955 0.647308
(0.06298) (0.23332) (0.35267) (0.06582) (0.05748)
5 7 1.65652 1.26586 0.984452 0.884972 0.680331
(0.01282) (0.21273) (0.47778) (0.0822) (0.086)
1 8 1.834814 1.336682 1.105877 0.969938 0.731993
(0.00872) (0.06445) (0.18032) (0.06854) (0.0305)
2 8 1.3445 1.098259 0.911702 0.786354 0.639154
(0.08542) (0.35259) (0.3425) (0.08098) (0.05902)
3 8 1.409338 1.144625 0.918137 0.809399 0.652189
(0.04877) (0.30648) (0.33359) (0.07725) (0.05295)
4 8 1.441858 1.180588 0.913955 0.836534 0.66258
(0.04922) (0.24257) (0.35083) (0.06924) (0.06739)
5 8 1.639143 1.256499 0.986119 0.889444 0.699727
(0.00798) (0.21802) (0.48027) (0.08) (0.09575)
1 9 1.838499 1.353782 1.10468 0.96316 0.737998
(0.00692) (0.05558) (0.20069) (0.06709) (0.03045)
2 9 1.327646 1.087341 0.910584 0.796562 0.644654
(0.09164) (0.3699) (0.3432) (0.08191) (0.05365)
3 9 1.4222 1.166569 0.950085 0.832056 0.689286
(0.04185) (0.2914) (0.39639) (0.07023) (0.05223)
4 9 1.441614 1.20656 0.949253 0.892197 0.687536
(0.04712) (0.22489) (0.40962) (0.0538) (0.052)
5 9 1.638964 1.246824 1.005401 0.925538 0.703552
(0.00792) (0.23507) (0.49254) (0.07909) (0.09752)
1 10 1.845398 1.333139 1.110443 0.956677 0.742753
(0.0058) (0.05793) (0.20838) (0.06726) (0.03102)
2 10 1.326481 1.07038 0.918588 0.809331 0.651274
(0.09575) (0.38947) (0.35247) (0.08257) (0.06164)
3 10 1.411594 1.16055 0.956158 0.840316 0.682552
(0.04143) (0.29351) (0.40946) (0.06768) (0.04674)
4 10 1.414903 1.202318 0.960569 0.898064 0.688243
(0.0469) (0.22498) (0.42869) (0.05585) (0.05286)
5 10 1.513267 1.21218 0.996095 0.940171 0.714841
(0.01402) (0.25008) (0.49448) (0.0835) (0.10901)
1 11 1.829338 1.32992 1.096103 0.947961 0.761543
(0.00452) (0.05778) (0.20241) (0.05946) (0.04182)
2 11 1.26995 1.031442 0.887507 0.789943 0.669607
(0.12552) (0.44864) (0.28599) (0.0714) (0.06795)
3 11 1.385748 1.120656 0.929876 0.83002 0.713875
(0.04195) (0.33931) (0.36197) (0.07206) (0.06534)
4 11 1.402516 1.144073 0.946521 0.887472 0.674693
(0.04838) (0.27993) (0.40517) (0.0577) (0.05353)
5 11 1.522067 1.231976 0.99724 0.945555 0.763958
(0.01432) (0.23051) (0.4961) (0.08173) (0.15832)
1 12 1.807784 1.322345 1.086972 0.940399 0.756602
(0.00541) (0.06401) (0.22226) (0.06526) (0.03608)
2 12 1.269219 1.034278 0.890161 0.826379 0.679187
(0.10345) (0.44712) (0.29618) (0.06412) (0.07191)
3 12 1.412307 1.116139 0.922254 0.806452 0.701781
(0.04053) (0.34262) (0.35429) (0.06921) (0.072)
4 12 1.451825 1.144524 0.949093 0.866915 0.705013
(0.0427) (0.27669) (0.4125) (0.05939) (0.08974)
5 12 1.554862 1.218466 0.996754 0.953275 0.770572
(0.01532) (0.24158) (0.49542) (0.08195) (0.18869)
Appendix C Inflation Forecast Using Rolling Window
Table 1
AO Random-Walk Model Number of quarters ahead
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
RMSE 0.020156 0.025432 0.028614 0.029474 0.024249
Table 2
Direct Autoregressive Model Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1.504254 1.187336 0.985107 0.847962 0.722946
(0.0984) (0.4395) (0.9851) (0.4913) (0.054)
2 1.611495 1.362431 1.06142 0.877392 0.729724
(0.1082) (0.3251) (0.5786) (0.5449) (0.0683)
3 1.747693 1.465887 1.118506 0.925926 0.738741
(0.0714) (0.1106) (0.6619) (0.4257) (0.0824)
4 1.788609 1.454975 1.086826 0.923337 0.737677
(0.0382) (0.0996) (0.76) (0.6038) (0.0448)
5 1.926681 1.522186 1.083033 0.890774 0.716768
(0.0356) (0.0979) (0.5262) (0.5543) (0.0345)
BIC 1.509214 1.31289 0.9872 0.827613 0.731004
(0.094) (0.4205) (0.9812) (0.5215) (0.0371)
Table 3
Multivariate Leading Index
Model
Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC 2.409065 2.376254 2.373136 1.650102 0.886556
(0.0026) (0.0012) (0.0077) (0.1765) (0.4334)
Table 4
Autoregressive Phillips Curve
Model
Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC 1.506789 1.367609 1.018308 0.853035 0.757024
(0.105) (0.4066) (0.8778) (0.5426) (0.0832)
Table 5
Factor Model Version (i) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
0 Bai-Ng 1.419428 0.9621 0.780893 0.724067 0.705992
Test
(0.127) (0.8835) (0.5776) (0.4266) (0.0663)
Table 6
Factor Model Version (ii) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC Bai-Ng
Test 1.68485 1.069491 0.811811 0.767061 0.863826
(0.101) (0.3326) (0.6081) (0.4755) (0.1324)
Table 7
Factor Model Version (iii) Number of periods ahead
No Lags Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1.404836 0.982712 0.814066 0.738956 0.69896
(0.0856) (0.9999) (0.6563) (0.4509) (0.0601)
2 1.423361 0.968681 0.786246 0.728238 0.745614
(0.1248) (0.9042) (0.5758) (0.4414) (0.1019)
3 1.406777 0.97838 0.79479 0.73346 0.748083
(0.1161) (0.9169) (0.5136) (0.4562) (0.111)
4 1.524566 1.037003 0.843927 0.7481 0.750203
(0.0603) (0.8827) (0.3722) (0.4778) (0.1011)
5 1.52282 1.048671 0.849994 0.750207 0.754543
(0.0837) (0.6388) (0.3537) (0.4827) (0.1209)
6 1.526981 1.046177 0.838813 0.750913 0.753073
(0.1146) (0.5977) (0.3485) (0.484) (0.1212)
7 1.578032 1.048308 0.841071 0.750523 0.762064
(0.0942) (0.6384) (0.3519) (0.4831) (0.1645)
8 1.625476 1.095425 0.851157 0.73652 0.763294
(0.0887) (0.6331) (0.3149) (0.4641) (0.173)
9 1.696446 1.103255 0.816431 0.73591 0.769487
(0.0588) (0.4619) (0.3329) (0.4476) (0.1848)
10 1.622881 1.049387 0.806861 0.737453 0.771141
(0.0632) (0.777) (0.334) (0.4572) (0.202)
11 1.633336 1.05375 0.797158 0.742469 0.773538
(0.0369) (0.7315) (0.3365) (0.4611) (0.1904)
12 1.673907 1.065819 0.790032 0.739957 0.770566
(0.0374) (0.7198) (0.3464) (0.464) (0.1968)
Table 8
Factor Model Version (iv) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1 1.664288 1.0591634 0.7840825 0.7065539 0.7084216
(0.0788) (0.3539) (0.2627) (0.3893) (0.082)
2 1 1.6708711 1.3000977 0.8179281 0.7193894 0.7107106
(0.0908) (0.0303) (0.4643) (0.4052) (0.0847)
3 1 1.6624034 1.3711399 0.8393779 0.7339669 0.7201649
(0.0847) (0.0639) (0.4314) (0.4228) (0.0709)
4 1 1.5518997 1.3481308 0.9662618 0.7411814 0.7449118
(0.1692) (0.2428) (0.1704) (0.4224) (0.2162)
5 1 1.5244552 1.3590925 0.9402105 0.8279667 0.7477064
(0.2007) (0.2028) (0.1437) (0.5276) (0.1801)
1 2 1.6727029 1.0085036 0.7175171 0.683594 0.7046139
(0.0786) (0.4653) (0.4028) (0.3206) (0.0858)
2 2 1.6597415 1.274406 0.7755491 0.7201252 0.6896425
(0.1057) (0.0285) (0.589) (0.3684) (0.0961)
3 2 1.6152632 1.3467682 0.7973234 0.7479187 0.7329178
(0.0593) (0.0392) (0.6082) (0.3892) (0.064)
4 2 1.5777658 1.2656731 0.8533148 0.7149149 0.7522441
(0.1449) (0.1723) (0.2879) (0.3604) (0.1631)
1 3 1.6616267 1.013657 0.7213038 0.6808354 0.7033741
(0.087) (0.4922) (0.4167) (0.3305) (0.0871)
2 3 1.6512158 1.2603874 0.7764475 0.7250455 0.6959007
(0.1088) (0.0159) (0.567) (0.3751) (0.1184)
3 3 1.6117472 1.3090342 0.8049873 0.7566724 0.7196803
(0.0769) (0.0595) (0.6004) (0.3934) (0.0654)
4 3 1.5904654 1.2549119 0.8382993 0.725785 0.730016
(0.1577) (0.1587) (0.4268) (0.3753) (0.1113)
5 3 1.4724833 1.2414227 0.8195159 0.7873672 0.7497358
(0.2342) (0.1559) (0.5142) (0.4779) (0.0451)
1 4 1.6119523 1.031442 0.7721815 0.690988 0.7062401
(0.0885) (0.3609) (0.294) (0.3517) (0.0992)
2 4 1.6260352 1.2437603 0.7880434 0.7274655 0.70361
(0.122) (0.0449) (0.5763) (0.3992) (0.126)
3 4 1.6097403 1.2631681 0.8265464 0.7570313 0.7185116
(0.0855) (0.0872) (0.6273) (0.396) (0.0645)
4 4 1.5581422 1.2441897 0.8343237 0.7227245 0.7443359
(0.1717) (0.1726) (0.3763) (0.3766) (0.129)
5 4 1.4892932 1.2232594 0.8260941 0.8191448 0.7524759
(0.1984) (0.1666) (0.4268) (0.4827) (0.0842)
1 5 1.6176787 1.0223384 0.7660188 0.6848379 0.7122787
(0.0817) (0.3639) (0.2938) (0.3265) (0.1175)
2 5 1.6461132 1.2390004 0.7935509 0.7322707 0.7068637
(0.1096) (0.0451) (0.5846) (0.4223) (0.1487)
3 5 1.6338109 1.3109025 0.8236 0.7530897 0.744241
(0.064) (0.1301) (0.602) (0.3876) (0.2002)
4 5 1.5817668 1.2770376 0.8286575 0.7106038 0.7659614
(0.1444) (0.2469) (0.362) (0.3578) (0.2484)
5 5 1.5799188 1.2111757 0.8396522 0.8091232 0.7450806
(0.1049) (0.1428) (0.2469) (0.4636) (0.0366)
1 6 1.626887 1.0385314 0.7638195 0.6844855 0.7118859
(0.0658) (0.3282) (0.2818) (0.3232) (0.1144)
2 6 1.6494443 1.2336846 0.787 0.7376929 0.7137075
(0.1122) (0.0449) (0.5889) (0.4482) (0.1655)
3 6 1.6341951 1.3348846 0.8254147 0.7602153 0.7487247
(0.0715) (0.1132) (0.5829) (0.4) (0.1828)
4 6 1.586449 1.2944178 0.8347329 0.722487 0.7695747
(0.1238) (0.2365) (0.4259) (0.3813) (0.229)
5 6 1.6017094 1.2132155 0.8547078 0.79716 0.7317243
(0.0898) (0.1458) (0.1621) (0.481) (0.0449)
1 7 1.6233038 1.0478504 0.7544493 0.6901983 0.7073024
(0.073) (0.3366) (0.3021) (0.3459) (0.1284)
2 7 1.65283 1.2357587 0.7790967 0.7588737 0.7051132
(0.11) (0.0433) (0.5525) (0.4518) (0.1713)
3 7 1.6364275 1.3683469 0.848946 0.7892238 0.7332086
(0.074) (0.1476) (0.5405) (0.4228) (0.179)
4 7 1.5992606 1.2812232 0.8419107 0.7282023 0.7856665
(0.121) (0.2138) (0.5021) (0.3982) (0.2366)
5 7 1.619894 1.2751191 0.8533011 0.7960924 0.7332989
(0.0684) (0.1304) (0.1854) (0.4883) (0.0208)
1 8 1.6550269 1.0422835 0.7290622 0.6969537 0.7084025
(0.0567) (0.4681) (0.4224) (0.3867) (0.1088)
2 8 1.6319252 1.2117092 0.7764776 0.7758653 0.718633
(0.0992) (0.0901) (0.555) (0.4717) (0.0826)
3 8 1.5952712 1.3867556 0.8212589 0.8230702 0.7302381
(0.0932) (0.1393) (0.5945) (0.4568) (0.1935)
4 8 1.573993 1.3601346 0.8346759 0.7682195 0.8335882
(0.1033) (0.2176) (0.4628) (0.4301) (0.038)
5 8 1.7099923 1.3345994 0.826007 0.7964027 0.7935639
(0.0753) (0.0849) (0.4908) (0.5647) (0.3256)
1 9 1.6259966 1.0380794 0.7145935 0.688937 0.7505045
(0.0475) (0.5604) (0.4282) (0.3629) (0.0877)
2 9 1.6504422 1.2492783 0.804965 0.7637004 0.7488072
(0.0853) (0.0278) (0.5546) (0.4295) (0.0929)
3 9 1.6342375 1.3715148 0.8487656 0.8312616 0.7465864
(0.0925) (0.1978) (0.5094) (0.4991) (0.2119)
4 9 1.6187274 1.4101191 0.9537172 0.7778448 0.8393009
(0.0871) (0.3393) (0.1229) (0.425) (0.0413)
5 9 1.7426157 1.461582 0.8926487 0.8439849 0.7673679
(0.0657) (0.1448) (0.09) (0.5806) (0.2629)
1 10 1.5491352 1.0325499 0.7146293 0.691431 0.7731506
(0.0456) (0.5388) (0.42) (0.3693) (0.0592)
2 10 1.6502905 1.228216 0.832613 0.7801035 0.7949067
(0.0635) (0.0391) (0.5194) (0.4751) (0.085)
3 10 1.6436548 1.4643579 0.830816 0.8552365 0.8095547
(0.0869) (0.3126) (0.5357) (0.4927) (0.1442)
4 10 1.753154 1.4407767 0.9188797 0.7712108 0.9058
(0.036) (0.4205) (0.1666) (0.4173) (0.0967)
5 10 1.7678161 1.3867061 0.9239108 0.8124478 0.8503855
(0.0537) (0.1457) (0.9304) (0.5963) (0.1578)
1 11 1.5565231 1.043585 0.7304528 0.680885 0.8077332
(0.0413) (0.8266) (0.4208) (0.3432) (0.0561)
2 11 1.6593492 1.1353453 0.8400311 0.8002724 0.8212896
(0.0795) (0.2544) (0.452) (0.5012) (0.0671)
3 11 1.643109 1.4795633 0.9218304 0.9882717 0.8105468
(0.0953) (0.295) (0.1399) (0.536) (0.1444)
4 11 1.7557473 1.4062244 0.866866 0.7629437 0.9298047
(0.0259) (0.3579) (0.281) (0.3809) (0.1215)
5 11 1.8142915 1.2379518 0.8991062 0.8506304 0.9818662
(0.0257) (0.3242) (0.93) (0.6922) (0.0362)
1 12 1.5480611 1.0474744 0.7438101 0.6822789 0.8209919
(0.0676) (0.9277) (0.4219) (0.3605) (0.0632)
2 12 1.6944992 1.1588999 0.8963043 0.8039896 0.7708797
(0.0727) (0.2175) (0.2045) (0.5002) (0.0755)
3 12 1.6741854 1.3099761 0.9537163 0.9488489 0.7738955
(0.0746) (0.1927) (0.1219) (0.6575) (0.1548)
4 12 1.7010796 1.3645506 0.8935598 0.7620995 1.0608415
(0.0371) (0.3253) (0.1868) (0.4498) (0.2301)
5 12 1.8527262 1.2640211 1.1389237 0.9625845 1.1424182
(0.0386) (0.0682) (0.4515) (0.085) (0.0299)
Appendix D IPI Growth Forecast Using Fixed Window
Table 1
AO Random-Walk Model Number of quarters ahead
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
RMSE 0.111915 0.122462 0.122077 0.114456 0.135254
Table 2
Direct Autoregressive Model Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 2.318737 1.250783 0.915867 0.851994 0.511659
(0.00075) (0.04791) (0.32597) (0.14298) (0)
2 2.372857 1.294724 0.978726 0.891125 0.517688
(0.00082) (0.04979) (0.46432) (0.23095) (0)
3 2.468914 1.245273 0.969661 0.895223 0.526134
(0.00113) (0.11866) (0.44129) (0.23417) (0.00001)
4 2.431552 1.246779 0.988979 0.903573 0.531885
(0.00634) (0.10185) (0.47863) (0.25499) (0.00003)
5 2.492654 1.212785 0.976159 0.916037 0.527053
(0.00674) (0.12589) (0.45376) (0.27948) (0.00009)
BIC 2.343673 1.287583 0.979833 0.867206 0.52839
(0.00062) (0.05114) (0.46236) (0.17432) (0)
Table 3
Factor Model Version (i) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
0 Bai-Ng
Test 2.302779 1.215523 0.995767 0.907495 0.529224
(0.01444) (0.05966) (0.49083) (0.2726) (0)
Table 4
Factor Model Version (ii) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC Bai-Ng
Test 2.424432 1.243039 1.033486 0.963581 0.539125
(0.01727) (0.12766) (0.4409) (0.41265) (0)
Table 5
Factor Model Version (iii) Number of periods ahead
No Lags Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 2.287181 1.272762 0.928425 0.842712 0.517968
(0.00404) (0.05994) (0.33085) (0.13131) (0)
2 2.325984 1.194515 0.974214 0.933081 0.536016
(0.01232) (0.09811) (0.44615) (0.33792) (0)
3 2.387296 1.172138 0.949354 0.939366 0.541492
(0.0114) (0.12538) (0.40368) (0.36009) (0)
4 2.39316 1.185536 0.945285 0.940848 0.530139
(0.00848) (0.10111) (0.3931) (0.37165) (0)
5 2.409928 1.184391 0.98373 0.967172 0.539584
(0.01012) (0.12566) (0.47151) (0.43148) (0)
6 2.478329 1.287979 1.10941 0.962375 0.549796
(0.00781) (0.08357) (0.35824) (0.41876) (0)
7 2.507593 1.320938 1.121481 0.950383 0.548514
(0.00622) (0.06193) (0.34138) (0.39189) (0)
8 2.565134 1.287052 1.154317 0.953755 0.553481
(0.00507) (0.0712) (0.30471) (0.39612) (0)
9 2.579179 1.29242 1.163307 0.997267 0.568814
(0.00463) (0.08112) (0.29723) (0.4934) (0)
10 2.513438 1.275846 1.160537 0.986032 0.583943
(0.00594) (0.08863) (0.30234) (0.46539) (0)
11 2.47019 1.229356 1.161593 0.976007 0.584522
(0.00545) (0.12324) (0.29134) (0.43715) (0)
12 2.392946 1.197087 1.148253 0.987234 0.596156
(0.00749) (0.17254) (0.30818) (0.46471) (0)
Table 6
Factor Model Version (iv) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1 2.478061 1.223505 0.923842 0.857433 0.538753
(0.00102) (0.03804) (0.33535) (0.15654) (0)
2 1 2.301786 1.340806 1.055756 0.924388 0.518524
(0.00214) (0.04852) (0.40482) (0.30641) (0)
3 1 2.35415 1.294461 1.045017 0.927762 0.525642
(0.0052) (0.11423) (0.41053) (0.30453) (0.00001)
4 1 2.276635 1.294235 1.067356 0.939 0.527259
(0.01819) (0.08966) (0.36319) (0.32647) (0.00002)
5 1 2.310526 1.272 1.055497 0.947115 0.534071
(0.01633) (0.09461) (0.38832) (0.34534) (0.00011)
1 2 2.621768 1.107334 0.883275 0.834041 0.514245
(0.00162) (0.23493) (0.23218) (0.13155) (0)
2 2 2.286481 1.272515 1.047898 0.851512 0.518397
(0.00121) (0.05727) (0.41307) (0.13716) (0)
3 2 2.336144 1.242893 1.009306 0.862111 0.519643
(0.01638) (0.14423) (0.47917) (0.13572) (0)
4 2 2.351261 1.134828 1.019021 0.841992 0.533429
(0.02512) (0.22238) (0.45938) (0.1149) (0.00001)
1 3 2.605994 1.077391 0.850445 0.806686 0.526945
(0.00129) (0.30563) (0.20171) (0.10637) (0)
2 3 2.329769 1.298109 1.036471 0.838384 0.518192
(0.00092) (0.02961) (0.43163) (0.12972) (0)
3 3 2.348262 1.215425 0.976505 0.85764 0.53488
(0.01329) (0.17959) (0.45546) (0.15498) (0)
4 3 2.347628 1.12694 0.980631 0.835392 0.528396
(0.02292) (0.23886) (0.46582) (0.13287) (0.00001)
5 3 2.369906 1.416224 1.121372 1.004458 0.528144
(0.00314) (0.04481) (0.31162) (0.49059) (0.00001)
1 4 2.786979 1.062777 0.853823 0.826876 0.527468
(0.0056) (0.34499) (0.20287) (0.12333) (0)
2 4 2.345071 1.289454 1.046257 0.893557 0.521153
(0.00097) (0.02554) (0.41358) (0.1691) (0)
3 4 2.407148 1.172457 0.978769 0.957884 0.54912
(0.01394) (0.22269) (0.45931) (0.3379) (0)
4 4 2.404322 1.118394 1.010417 0.940063 0.546658
(0.01957) (0.26024) (0.48215) (0.2733) (0)
5 4 2.499153 1.484022 1.169309 1.090768 0.537657
(0.00168) (0.07227) (0.26198) (0.27524) (0.00003)
1 5 2.83388 1.15685 0.920973 0.827054 0.542398
(0.00146) (0.16586) (0.32257) (0.11906) (0)
2 5 2.433406 1.358623 1.163381 0.967385 0.529797
(0.00095) (0.01114) (0.24815) (0.36486) (0)
3 5 2.414455 1.203936 1.045435 1.004355 0.549126
(0.01582) (0.20961) (0.42845) (0.48379) (0)
4 5 2.431403 1.168887 1.120133 1.013361 0.552695
(0.01967) (0.20394) (0.3235) (0.45111) (0.00001)
5 5 2.469536 1.485984 1.205492 1.138157 0.540677
(0.00205) (0.07171) (0.18995) (0.20906) (0.00004)
1 6 2.833099 1.142383 0.888015 0.826422 0.552408
(0.00058) (0.17027) (0.23867) (0.13558) (0)
2 6 2.433338 1.472256 1.245753 1.052597 0.540934
(0.00162) (0.0182) (0.2236) (0.36227) (0)
3 6 2.382037 1.222231 0.969702 0.981753 0.556535
(0.02233) (0.19594) (0.43827) (0.43173) (0)
4 6 2.407493 1.276729 1.221349 1.091277 0.549313
(0.02583) (0.11726) (0.24288) (0.28821) (0.00001)
5 6 2.290308 1.529576 1.352706 1.240056 0.541292
(0.00264) (0.04851) (0.11912) (0.15808) (0.00008)
1 7 2.810144 1.182384 0.980939 0.931486 0.552078
(0.00054) (0.17494) (0.45365) (0.31302) (0)
2 7 2.398785 1.544477 1.26958 1.03604 0.558626
(0.00184) (0.01053) (0.21118) (0.41024) (0)
3 7 2.358812 1.243257 0.94685 1.042888 0.585946
(0.02416) (0.15577) (0.4036) (0.33162) (0)
4 7 2.42457 1.271416 1.235693 1.126368 0.574982
(0.02328) (0.10861) (0.24025) (0.22137) (0.00001)
5 7 2.330913 1.52611 1.32495 1.247734 0.589763
(0.002) (0.05017) (0.15295) (0.14309) (0.00083)
1 8 2.788621 1.210109 0.953807 0.914599 0.529047
(0.0005) (0.14929) (0.37742) (0.28962) (0)
2 8 2.474605 1.532041 1.30222 1.072283 0.532019
(0.00215) (0.01793) (0.18782) (0.32385) (0.00001)
3 8 2.337216 1.220633 0.926836 1.039626 0.571872
(0.02571) (0.18238) (0.36854) (0.32712) (0)
4 8 2.456241 1.243432 1.265924 1.112036 0.603164
(0.02101) (0.12932) (0.22798) (0.26704) (0.00018)
5 8 2.36141 1.609896 1.389605 1.250634 0.585077
(0.00122) (0.04051) (0.10495) (0.14304) (0.00064)
1 9 2.647032 1.232546 0.942922 0.917451 0.532668
(0.0001) (0.12703) (0.34564) (0.31097) (0)
2 9 2.445711 1.538123 1.304356 1.079229 0.53738
(0.0023) (0.01719) (0.18453) (0.28833) (0.00002)
3 9 2.341157 1.238356 0.941302 1.061719 0.586321
(0.02961) (0.1666) (0.38634) (0.21644) (0.00001)
4 9 2.440484 1.259386 1.246711 1.107949 0.596186
(0.02259) (0.11639) (0.22665) (0.26056) (0.00011)
5 9 2.325231 1.619675 1.40009 1.225651 0.588048
(0.00145) (0.02993) (0.09774) (0.13922) (0.00124)
1 10 2.648386 1.246474 0.936903 0.943792 0.55825
(0.00009) (0.1127) (0.32919) (0.36246) (0)
2 10 2.469031 1.53897 1.339495 1.075452 0.558784
(0.00177) (0.02274) (0.16151) (0.30758) (0)
3 10 2.331697 1.222462 0.947684 1.086824 0.590097
(0.02902) (0.1827) (0.40518) (0.18727) (0.00005)
4 10 2.423908 1.310085 1.225485 1.150883 0.598126
(0.01951) (0.0789) (0.25067) (0.19539) (0.00002)
5 10 2.275641 1.645929 1.46983 1.257995 0.589966
(0.00098) (0.02904) (0.06688) (0.09715) (0.00226)
1 11 2.636645 1.278185 0.972214 0.940008 0.542394
(0.00008) (0.06743) (0.42041) (0.37366) (0)
2 11 2.500471 1.500899 1.328236 1.073707 0.554438
(0.0016) (0.03391) (0.17037) (0.31529) (0)
3 11 2.363829 1.224427 0.947619 1.087786 0.592821
(0.02248) (0.18204) (0.40667) (0.1697) (0.00007)
4 11 2.352904 1.288209 1.230561 1.157903 0.650998
(0.0157) (0.11549) (0.24028) (0.1909) (0)
5 11 2.260823 1.649149 1.451874 1.251602 0.619856
(0.00096) (0.04132) (0.06896) (0.06243) (0.00325)
1 12 2.61764 1.26897 0.958413 0.901644 0.551818
(0.00002) (0.07314) (0.36418) (0.3161) (0)
2 12 2.556828 1.537699 1.319531 1.018978 0.562136
(0.00185) (0.01518) (0.16168) (0.45736) (0)
3 12 2.332926 1.228231 0.939502 1.110238 0.61686
(0.02381) (0.19643) (0.3949) (0.11741) (0.00001)
4 12 2.330109 1.271854 1.215345 1.105987 0.670404
(0.01102) (0.12825) (0.25788) (0.3119) (0)
5 12 2.279401 1.638254 1.454004 1.219 0.627205
(0.0007) (0.03996) (0.0555) (0.11955) (0.00609)
Appendix E IPI Growth Forecast Using Rolling Window
Table 1
AO Random-Walk Model Number of quarters ahead
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
RMSE 0.111915 0.122462 0.122077 0.114456 0.135254
Table 2
Direct Autoregressive Model Number of periods ahead
Number of
lags
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1
2.30422 1.213299 0.897886 0.841357 0.523612
(0.00098) (0.21739) (0.42498) (0.42287) (0.06327)
2
2.398289 1.289219 0.978382 0.91639 0.54173
(0.0009) (0.15725) (0.28062) (0.47615) (0.01739)
3
2.601302 1.274025 1.053907 0.95177 0.569741
(0.00235) (0.01996) (0.40844) (0.94362) (0.16122)
4
2.622283 1.339035 1.078532 0.978107 0.582727
(0.00018) (0.03664) (0.23102) (0.68791) (0.39225)
5
2.847081 1.409442 1.130601 1.073264 0.618165
(0.11346) (0.41151) (0.62008) (0.48687) (0.16876)
BIC
2.353 1.341341 1.088674 0.986252 0.621465
(0.07896) (0.18029) (0.65939) (0.37967) (0.12454)
Table 3
Factor Model Version (i) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
0 Bai-Ng
Test 2.377937 1.2967 0.851299 0.773992 0.544998
(0.07896) (0.18029) (0.65939) (0.37967) (0.12454)
Table 4
Factor Model Version (ii) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
BIC Bai-Ng
Test 3.440653 1.891393 1.492572 1.023374 0.640193
(0.01287) (0.01253) (0.13583) (0.1726) (0.06106)
Table 5
Factor Model Version (iii) Number of periods ahead
No Lags Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 2.25671 1.226911 0.916234 0.850441 0.563657
(0.04603) (0.04871) (0.92269) (0.81022) (0.13156)
2 2.402692 1.344415 0.966312 0.879066 0.522694
(0.07082) (0.08143) (0.89746) (0.36132) (0.13033)
3 2.427535 1.397815 0.993428 0.885498 0.525647
(0.06687) (0.0991) (0.77099) (0.50652) (0.12836)
4 2.561374 1.433352 0.914335 0.82002 0.516027
(0.11943) (0.50555) (0.87385) (0.3335) (0.09659)
5 2.54497 1.53438 0.916907 0.812243 0.511202
(0.07854) (0.48528) (0.70696) (0.49997) (0.08239)
6 2.60909 1.550732 0.95316 0.819891 0.49813
(0.05217) (0.42099) (0.65434) (0.44564) (0.05661)
7 2.623027 1.558043 0.991293 0.85061 0.520029
(0.05106) (0.41747) (0.71209) (0.47009) (0.06541)
8 2.509469 1.585167 1.007555 0.853542 0.512395
(0.06474) (0.35484) (0.81764) (0.5531) (0.06935)
9 2.519252 1.613203 1.004842 0.871251 0.519044
(0.04884) (0.33489) (0.79277) (0.42742) (0.12359)
10 2.563141 1.597731 1.00028 0.87415 0.524453
(0.04894) (0.33089) (0.77925) (0.81577) (0.12054)
11 2.678076 1.659914 0.985833 0.873141 0.524988
(0.05701) (0.22812) (0.93322) (0.94697) (0.12346)
12 2.705526 1.715351 0.958331 0.86296 0.514536
(0.05143) (0.19569) (0.87694) (0.91799) (0.11862)
Table 6
Factor Model Version (iv) Number of periods ahead
Number of
lags
Number of
Factors
1 quarter 2 quarter 3 quarter 4 quarter 8 quarter
1 1 3.398468 1.767145 1.285923 0.969059 0.56387
(0.01165) (0.00572) (0.26597) (0.15348) (0.05029)
2 1 3.377153 1.894319 1.58778 1.139547 0.603002
(0.01075) (0.18872) (0.0206) (0.50998) (0.0198)
3 1 3.322196 1.77861 1.827876 1.347516 0.622243
(0.0137) (0.06615) (0.10405) (0.01592) (0.0172)
4 1 3.368682 1.770439 1.836798 1.388301 0.620682
(0.01735) (0.08197) (0.10517) (0.00092) (0.02299)
5 1 3.508041 1.895216 1.874205 1.361988 0.633173
(0.01644) (0.18189) (0.16033) (0.00382) (0.00073)
1 2 3.417657 1.740823 1.187546 0.912986 0.588337
(0.01298) (0.02013) (0.31233) (0.84341) (0.0597)
2 2 3.299513 1.901604 1.5666 1.040551 0.658555
(0.00858) (0.20439) (0.06557) (0.6681) (0.03481)
3 2 3.304853 1.764427 1.784419 1.305226 0.626005
(0.01553) (0.06484) (0.07464) (0.0405) (0.01586)
4 2 3.350515 1.715954 1.760072 1.321891 0.660471
(0.01575) (0.06692) (0.14678) (0.00563) (0.01356)
1 3 3.415567 1.687621 1.173445 0.922708 0.588768
(0.01279) (0.00624) (0.20054) (0.78816) (0.05653)
2 3 3.327388 1.86556 1.488291 1.026808 0.666754
(0.00905) (0.16913) (0.01593) (0.694) (0.04036)
3 3 3.335011 1.730117 1.73623 1.306436 0.617303
(0.01682) (0.05557) (0.03177) (0.04116) (0.01232)
4 3 3.333733 1.69643 1.716658 1.331709 0.66959
(0.01533) (0.0709) (0.09302) (0.00608) (0.01771)
5 3 3.422678 1.831183 1.718526 1.292785 0.6307
(0.02047) (0.22727) (0.12758) (0.00085) (0.00049)
1 4 3.330663 1.69052 1.133567 0.861728 0.588054
(0.01202) (0.10613) (0.18427) (0.63472) (0.09692)
2 4 3.241797 1.859774 1.462272 1.051177 0.680919
(0.0092) (0.1879) (0.0325) (0.64419) (0.0466)
3 4 3.353818 1.697369 1.723995 1.316225 0.608628
(0.0178) (0.07158) (0.02008) (0.05932) (0.01733)
4 4 3.336173 1.671332 1.687111 1.349955 0.64611
(0.01628) (0.07674) (0.08179) (0.00037) (0.01737)
5 4 3.471196 1.785406 1.673688 1.263932 0.631525
(0.02898) (0.24246) (0.04131) (0.001) (0.00035)
1 5 3.377314 1.747319 1.140624 0.833806 0.582478
(0.01448) (0.08452) (0.26463) (0.69649) (0.0914)
2 5 3.275017 1.881975 1.487444 1.056989 0.67582
(0.01177) (0.18354) (0.02918) (0.64144) (0.04519)
3 5 3.453306 1.705846 1.710288 1.278094 0.624109
(0.02494) (0.08037) (0.01966) (0.05279) (0.01737)
4 5 3.356484 1.645424 1.681429 1.377558 0.639372
(0.01811) (0.06955) (0.09979) (0.0007) (0.02238)
5 5 3.408254 1.778833 1.621173 1.260435 0.664444
(0.02601) (0.2501) (0.02751) (0.00197) (0.01711)
1 6 3.360553 1.787002 1.145664 0.826015 0.591188
(0.01329) (0.06973) (0.18391) (0.67778) (0.1164)
2 6 3.295756 1.887435 1.523579 1.032944 0.671909
(0.0115) (0.1933) (0.01825) (0.55158) (0.07199)
3 6 3.444527 1.669846 1.759532 1.343427 0.630684
(0.02407) (0.09497) (0.02027) (0.06466) (0.0188)
4 6 3.35236 1.633397 1.791659 1.440072 0.650909
(0.01682) (0.07853) (0.09177) (0.00005) (0.04698)
5 6 3.381405 1.755551 1.704264 1.355118 0.676776
(0.02897) (0.24235) (0.02776) (0.00158) (0.0544)
1 7 3.327608 1.772593 1.13697 0.851707 0.600592
(0.01438) (0.09464) (0.17099) (0.54424) (0.11877)
2 7 3.296199 1.896844 1.519608 1.011571 0.668279
(0.01053) (0.20379) (0.02839) (0.55626) (0.06585)
3 7 3.421647 1.693513 1.750376 1.366509 0.684406
(0.0247) (0.10687) (0.01819) (0.0315) (0.00316)
4 7 3.32183 1.626138 1.799707 1.443522 0.710009
(0.01856) (0.07317) (0.06924) (0.0005) (0.01401)
5 7 3.349533 1.718345 1.760989 1.291031 0.686701
(0.02854) (0.20778) (0.08018) (0.00052) (0.03491)
1 8 3.351487 1.789504 1.155755 0.855412 0.623446
(0.01395) (0.111) (0.4391) (0.47366) (0.12376)
2 8 3.297842 1.900665 1.52663 1.046275 0.728309
(0.00955) (0.20101) (0.03675) (0.48497) (0.04359)
3 8 3.455872 1.695137 1.745275 1.364723 0.708788
(0.02466) (0.09954) (0.01647) (0.08307) (0.00507)
4 8 3.338161 1.609847 1.758501 1.400536 0.698676
(0.02475) (0.08781) (0.05629) (0.00189) (0.01473)
5 8 3.342007 1.788708 1.746579 1.282177 0.703475
(0.03398) (0.20953) (0.02987) (0.00096) (0.01498)
1 9 3.374867 1.773157 1.173255 0.834477 0.636764
(0.01298) (0.09411) (0.42018) (0.38021) (0.14797)
2 9 3.326817 1.894498 1.506731 1.044915 0.724331
(0.00901) (0.18514) (0.08369) (0.47639) (0.04417)
3 9 3.446655 1.688704 1.705608 1.357751 0.688655
(0.02619) (0.03905) (0.06657) (0.06566) (0.00656)
4 9 3.334194 1.606997 1.712715 1.385993 0.712224
(0.02455) (0.06899) (0.05194) (0.01513) (0.01789)
5 9 3.343078 1.760246 1.625537 1.278929 0.780544
(0.03633) (0.17253) (0.0308) (0.00036) (0.02647)
1 10 3.353857 1.678615 1.160262 0.829761 0.647014
(0.01262) (0.09641) (0.51237) (0.36162) (0.14898)
2 10 3.326697 1.901321 1.522625 1.014286 0.740106
(0.0091) (0.18607) (0.08443) (0.56657) (0.03989)
3 10 3.464704 1.718208 1.67932 1.389566 0.691157
(0.02521) (0.04615) (0.06636) (0.01244) (0.00417)
4 10 3.317848 1.628034 1.696254 1.359786 0.706187
(0.02532) (0.08181) (0.0445) (0.00663) (0.02357)
5 10 3.332849 1.747871 1.634155 1.283324 0.7186
(0.03934) (0.19684) (0.02021) (0.00004) (0.03277)
1 11 3.364461 1.730072 1.178409 0.826434 0.656508
(0.01333) (0.05548) (0.35556) (0.26726) (0.14154)
2 11 3.296029 1.951656 1.564787 1.097152 0.734515
(0.00939) (0.20082) (0.10787) (0.0462) (0.03285)
3 11 3.458068 1.780519 1.628312 1.398343 0.693872
(0.02602) (0.04098) (0.05171) (0.00438) (0.0077)
4 11 3.287124 1.568681 1.70696 1.359834 0.722983
(0.0253) (0.08401) (0.0435) (0.0088) (0.02276)
5 11 3.252039 1.703218 1.708251 1.373014 0.826061
(0.03424) (0.16888) (0.01671) (0.01162) (0.13632)
1 12 3.366698 1.816795 1.223046 0.835183 0.654286
(0.01323) (0.0389) (0.45309) (0.23639) (0.13644)
2 12 3.238194 2.028269 1.498884 1.126216 0.737504
(0.00928) (0.16359) (0.08153) (0.01097) (0.05389)
3 12 3.421456 1.693079 1.572251 1.365439 0.698172
(0.02419) (0.08087) (0.02463) (0.00883) (0.00719)
4 12 3.279827 1.549486 1.683885 1.334704 0.781332
(0.02248) (0.10798) (0.05664) (0.02843) (0.03032)
5 12 3.184654 1.793619 1.718191 1.500431 0.88414
(0.03052) (0.12904) (0.01238) (0.00036) (0.40221)