# 헤지펀드 수익률과 총요소생산성

# Hedge Fund Returns and Total Factor Productivity

## Article information

## Abstract

본 연구는 구조적 벡터자기회귀 모형(SVAR)과 벡터오차수정 모형(VECM)을 이용하여 헤지펀드들의 투자 행태가 거시 경제적 총요소생산성의 변화를 예측할 수 있는지 살펴보았다. 즉, 주식 시장의 정보보유 거래자(informed traders)로써 헤지펀드는 다른 투자자들보다 시장 분석에 있어서 더 우월한 능력을 가지고 있다고 알려져 있으며, 이에 헤지펀드들이 거시 경제적 생산성의 향상과 관련된 뉴스 충격을 선제적으로 포착하고 대응할 수 있는지 확인하였다. John Fernald(2014)가 산출한 미국의 분기별 TFP 자료와 크레딧 스위스/트레몬트 데이터베이스에서 제공하는 헤지펀드 지수(HFI)의 수익률 자료를 활용하여 1994년 1분기부터 2023년 2분기까지의 표본 기간 동안 두 내생 변수 간의 시계열적 연관성을 검토하였다. 분석기간 두 변수의 동 분기 상관계수는 0.9791로 나타나 두 변수의 움직임에 유사성이 매우 높다는 것이 확인되었다. 그랜저 인과관계 검정 결과, “Δ*Ln*(*HFI*)가 Δ*TFP*를 그랜저 인과하지 않는다”라는 가설이 기각되었으며 이는 헤지펀드들의 행태를 보여주는 헤지펀드 수익률 정보가 총요소생산성의 변화를 예측하는 데 유용하다는 것을 암시한다. 마지막으로, VECM 모델을 사용한 예측오차 분산분해에 따르면 Δ*Ln*(*HFI*)에 대한 충격이 20분기 이후에도 Δ*TFP*변동의 65% 이상을 설명하는 것으로 나타났다.

## Trans Abstract

This study explores whether hedge funds’ investment behavior can predict variations in productivity levels using a structural vector autoregressive model (SVAR) and a vector error correction model (VECM). As *informed* traders in the stock market with superior skills, hedge funds may quickly capture news shocks regarding future production growth in advance. With quarterly series of *TFP* provided by John Fernald (2014) and hedge fund index (*HFI*) returns obtained from the Credit Suisse/Tremont database, I find a contemporaneous correlation coefficient of 0.9791 between two endogenous variables over the sample period from 1Q:1994 to 2Q:2023, indicating a high degree of similarity in their movements. A Granger Causality Test rejects the hypothesis, “Δ*Ln*(*HFI*)does not Granger Cause Δ*TFP*”, suggesting that the information inferred from the hedge fund index are valuable in predicting future economic productivity. Finally, the forecast error variance decompositions using the VECM model indicate that over 65% of the variation in Δ*TFP* even after 20 quarters can be attributed to a shock to the Δ*Ln*(*HFI*).

**Keywords:**헤지펀드; 정보보유 거래자; 미국 총요소생산성; 구조적 벡터자기회귀 모형; 벡터오차수정 모형

**Keywords:**

**Keywords:**Hedge Funds; Informed Traders; U.S. Total Factor Productivity; SVAR; VECM

## 1. Introduction

According to Fama (1990), stock market return variations are fairly correlated with the growth rate of future production since both may reflect information about shocks to expected cash flows. Beaudry and Portier (2006) argue that stock price movements incorporate the expectations of economic agents regarding future economic conditions. Given that changes in agents’ expectations are related to future economic conditions, I posit that news shocks inferred from current stock price movements may explain future business cycle fluctuations. Therefore, I expect that investors participating in the stock market will move first, and then one can predict future economic conditions based on the behavior of the stock market participants.

To identify the relationship between stock market movements and future economic conditions, this study examines the cross-impacts between hedge fund index returns indicating the performance of hedge funds in aggregate and macroeconomic productivity using a structural vector autoregressive (SVAR) model and a vector error correction model (VECM). I assume that the investment behavior of hedge funds, as *informed* traders in the stock market, captures news shocks about future production growth more quickly than the movement of the total stock market index, such as the S&P 500 index examined in Beaudry and Portier (2006).

The movement of the total stock market index reflects the behavior of all types of investors, whether informed or not, in response to material information. If the stock market exhibits some degree of informational inefficiency, then the behavior of *uninformed* investors (i.e., noise traders) may also contribute to delayed stock price responses. Therefore, the total stock market index, which reflects the expectations of all types of investors, is less appropriate than the hedge fund index for capturing news about future economic conditions. In other words, if hedge fund managers are well-informed and have superior predictive skills, then the hedge fund index (i.e., aggregating the behaviors of informed traders) can be a more effective proxy for capturing agents’ expectations regarding future production growth. Recent studies examining hedge fund managers’ performance demonstrate their superior ability to identify mispricing in the stock market and gain an informational advantage in trading using various sources (Agarwal et al., 2009; Ben-David et al., 2012; Gao and Huang, 2016; Huang and Jain, 2024).

Following Beaudry and Portier (2006), I test the bi-variate aspects between the aggregate hedge fund index returns and total factor productivity (*TFP*). *TFP* is the part of output that is unexplained by inputs for production, such as labor and capital. *TFP* represents technological efficiency and explains economic growth (Easterly and Levine, 2001). Many studies empirically estimate country- or industry-level TFPs, but only annual frequency measures have been proposed because of data limitations. However, Fernald (2014) provides historical quarterly series of *TFP* based on a growth-accounting database for the U.S. business sector. This study employs the Fernald’s quarterly *TFP* series to measure productivity.

Using quarterly series of *TFP* and hedge fund index returns, this study examines the macroeconomic dynamics between hedge funds’ behaviors and economic productivity. The results are as follows. First, the contemporaneous correlation coefficient between the two variables is 0.9791, meaning that their movements are highly similar over the sample period. Note that the correlation coefficient between *TFP* and a S&P500 Index is 0.6856.^{1)} Second, a Granger Causality Test rejects the hypothesis, Δ*Ln*(*HFI*)does not Granger Cause Δ*TFP*, implying that the information inferred from the behaviors of hedge funds seems to be useful to predict future economic productivity. Finally, the forecast error variance decompositions with the VECM model indicate that more than 65% of the variation in Δ*TFP* after 20 quarters can be attributed to a shock to the Δ*Ln*(*HFI*).

## 2. Model Specification

### 2.1 Structural Vector Autoregressive Model (SVAR) and Impulse Responses

Sims (1980) introduced a new class of econometric models, SVAR, in which identification focuses on the errors of the system rather than identifying the autoregressive coefficients (Lutkepohl and Kraetzig, 2004). The innovations of the original VAR model are orthogonalized with a Cholesky decomposition of the covariance matrix; thus, a recursive structure is imposed on the instantaneous relationships between the variables. The drawback of applying the Cholesky decomposition to obtain impulse responses is that choosing the ordering of variables may produce different shocks. Thus, I need to check the robustness of the impulse responses by mixing the ordering of the main variables unless any theory supports the specific recursive structure.

SVAR models are similar to the VAR model but have more constraints to identify parameters. For example, the identification of shocks in an SVAR is based on economic theory, which suggests that the effects of some shocks are zero in the long- or short-run. In other words, I need to impose certain restrictions on the model parameters to identify the SVAR model. In the model of Sims (1980), the triangular (or recursive) identification scheme implies that an additional shock to the second variable does not affect the first (the most exogeneous) variable in the same period. The SVAR model is as follows:

where *y _{t}* is a (

*K*×1) vector of the endogenous variables,

*x*contains exogeneous variables, and the underlying structural shocks are

_{t}*ε*~ (0,

_{t}*I*). Matrix A specifies the instantaneous relations among the variables in

_{K}*y*, then the reduced-form disturbances are u

_{t}_{t}=

*A*

^{-1}

*Bε*. Using a Cholesky decomposition of the covariance matrix Σ

_{t}*and*

_{u}*B*to be a lower triangular matrix Σ

*=*

_{u}*BB*’’, I obtain the process

*y*=

_{t}*Φ*u

_{o}*+Φ*

_{t}_{1}u

*+… = Ψ*

_{t−1}*+Ψ*

_{o}ε_{t}_{1}

*ε*+…,, where Ψ

_{t−1}*= Φ*

_{i}*. I employ a long-run restriction on the SVAR following Blanchard and Quah (1989) and Beaudry and Portier (2006). Blanchard and Quah (1989) identify supply (demand) shocks with persistent (transitory) effects on output. Then, the (1,2)-element of the long-run impact matrix*

_{i}BBeaudry and Portier (2006) also adopt the VECM with the long-run restriction suggested by Blanchard and Quah (1989) for identification with two differenced variables: stock prices and *TFP*. Following Beaudry and Portier (2006), the estimation strategies of this study are in two ways, SVAR and VECM. First, the cointegration test with the Johansen method (allowing a linear trend in data and exogeneous variables) to examine whether a cointegration relationship exists between *HFI* and *TFP* level variables indicates no cointegration relationship between the two variables. However, for the first differenced variables, the cointegration test shows a statistically significant cointegration equation between the differenced endogenous variables. Thus, I employ the SVAR with the differenced variables and impose a restriction on the long-run impact matrix to observe the impulse responses of the shocks. Second, I adopt the VECM with differenced endogenous variables without restrictions as a robustness test.

The bivariate system is expressed as *TFP* and *HFI*. As deterministic terms, a constant term and a linear time trend term are included; and the HP-detrended stock market index, *SNP500*, and the HP-detrended University of Michigan Consumer Sentiment Index, *UM_SENT*, are also included as exogenous variables in the SVAR to compare the results with Beaudry and Portier (2006).^{2)} Additionally, this allows me to identify the news shock inferred through the behavior of informed investors who expect better future production growth than the one observed by the behavior of all informed and uninformed stock market participants. I chose an optimal lag order of 1 for the main endogenous variables through the VAR lag-length test and include one-lagged detrended exogeneous variables.

### 2.2 Vector Error Correction Models

When two or more variables are cointegrated, their time-series observations move together as a pair, and then, a linear combination of the two series exists to form a stationary process, although the individual time-series are non-stationary (e.g., the first-difference stationary I(1) process). Many macroeconomic variables are non-stationary in their levels, but their first differences are stationary, or I(1), and are often cointegrated with other variables. For instance, the GDP and the stock market index in the levels are individually non-stationary I(1) processes. However, these two time-series can be tied together, and the linear combination of the GDP and the stock market index becomes a stationary process.

A common way to analyze non-stationary I(1) data is to take the first-differences of the variables to make stationary processes. However, this approach may lose important information if the two variables are cointegrated. In other words, a Vector Autoregressive (VAR) model with first-differenced data does not capture the long-run relationship between two cointegrated variables. The Vector Error Correction Model (VECM) could be appropriate to analyze the cointegrated time-series by adding a vector of lagged error-correction terms in the VAR equations. In the case of the two variables, these error correction terms are lagged residuals from the cointegrating regression between the two variables in levels. The terms indicate the prior disequilibrium deviated from the long-run relationship, in which those residuals would be zero. The multivariate VECM specifications are as follows (Cottrell and Lucchetti, 2016):

Consider a VAR of order *p* with a deterministic part given by *μ _{t}* (typically, a polynomial in time) and a unmodeled stochastic or exogenous part,

*x*. One can write the

_{t}*n*-variate process

*y*. as

_{t}The equation (2) can be expressed as

where *n*, cointegration occurs and Π can be written as *αβ*’. Then, *β* is called a cointegrating vector and *α* is called an adjustment vector. In this case, *y _{t}* is I(1), but the combination (i.e., an error-correction term)

*ec*=

_{t}*β’y*is I(0). In fact,

_{t}*ec*represents the deviation from the long-run equilibrium relationship among the variables and may not be zero. The equation (3) is expressed as follows:

_{t}If *β* is known, then *ec _{t-1}* =

*β’y*would be observable and all the remaining parameters could be estimated via OLS (ordinary least squares). For example, if there are two endogenous variables with one cointegration relation, then

_{t-1}In this study, the VECM for the bivariate system is expressed as *TFP* and *HFI* are described in <Table 1>. I include a constant and error correction terms as well as the HP-detrended stock market index, *SNP500*, and the HP-detrended University of Michigan Consumer Sentiment Index, *UM_SENT*, as exogenous variables in the VECM. I chose an optimal lag order of 1 for the main endogenous variables through the VAR lag-length test and include one-lagged detrended exogeneous variables.

## 3. Data Descriptions

### 3.1 Hedge Fund Industry

Since 1994, the hedge fund industry has grown rapidly and drawn much attention from academia and the financial industry. <Figure 1> plots the estimated total assets under management (AUM) of the hedge fund industry. Although there are no standardized central statistics estimating the overall size of the hedge fund industry, its total AUM have increased from approximately $118 billion in 1997 to $5.14 trillion at its peak as of 2^{nd} quarter of 2023, according to the Barclay Hedge Fund database.^{3)} The hedge fund industry fell significantly once in 2008, but the total AUM size has been continuously growing in recent days since the 2008 global financial crisis.

In general, hedge fund managers are known for having exceptional skills to predict overall economic growth because of their outstanding performance. They are usually more *informed* about future economic news than individual or noise investors. Therefore, the economic predictions by hedge fund managers can be considered more precise than those of other types of investors in the stock market. In addition, they are less regulated than other institutional investors (e.g., mutual fund) because many offshore centers provide business-friendly regulations for the hedge funds. Thus, the hedge fund index is a better proxy for the news shocks to future production or economic conditions than the stock market index, which reflects the expectations of all market participants.

Among the various hedge fund indices, the Credit Suisse Hedge Fund (CSHF) Indexes has released the benchmark performance summary on both the aggregated composite index and ten style-based sector indexes since January 1994.^{4)}^{,} As for the sectors of hedge funds, hedge funds generally notify their trading strategies or styles to investors in their prospectuses.^{5)} The CSHF Indexes separate funds into ten primary sectors based on their declared investment styles. For example, the global macro funds attempt to generate excess returns by making leveraged bets on price movements in equity, currency, interest rate, and commodity markets. The macro section of the name explains that managers use macroeconomic principles to identify arbitrage opportunities in asset prices.

### 3.2 Sample Construction and Summary Statistics

To examine these hypotheses, I use the Credit Suisse Hedge Fund (CSHF) index provided by the Credit Suisse/Tremont database and a quarterly *TFP* series obtained from the Fernald’s website. The Credit Suisse/Tremont database, which tracks more than 9,000 funds, is used to determine the Index Universe. This selection universe is defined as only the funds with a minimum of $50 million under management, a minimum one-year track record, and current audited financial statements.^{6)} The CSHF index is calculated with asset-weighting and rebalanced semi-annually. The index includes new funds using a rule-based construction method; therefore, it tracks at least the top 85% of the AUM in each of the 10 universe or strategy categories. The historical monthly indices from January 1994 to the present are available on the website.

Fernald (2014) provides a quarterly series on *TFP* for the U.S. business sector. Shocks to quarterly estimated TFP are regarded as an important factor driving business cycle fluctuations (Kydland and Prescott, 1982). Because the *TFP* series are only available at a quarterly frequency, I also use the CSHF index data from the end of each quarter. Therefore, the sample period for all the analyses is from the first quarter of 1994 to the second quarter of 2023 (118 observations).

<Table 1> lists the detailed definitions of the main endogenous and exogenous variables. For the bi-variate SVAR and VECM, total factor productivity and the hedge fund index, (*TFP _{t}* ,

*HFI*), are employed as endogenous variables and the HP-detrended University of Michigan Consumer Sentiment Index and the HP-detrended stock market index, (

_{t}*UM*_

*SENT*,

_{t}*SNP*500

*), are included as exogenous variables. Barsky and Sims (2012) emphasize the importance of consumer confidence in forecasting future economic activity as confidence may reflect agents’ expectations of future economic productivity. Therefore, both the stock market index and consumer confidence survey results could also closely relate to future economic conditions.*

_{t}<Figure 2> displays the time-series plots of the variables over the sample period from 1Q:1994 to 2Q:2023. For comparison with the productivity measures, <Figure 2> also reports the time-series plots of log-transformed utility-adjusted *TFP* series and business output series provided by Fernald (2014). Surprisingly, the hedge fund index (*HFI*) and log-transformed TFP series (*TFP*) show extremely similar patterns over the sample period. The correlation coefficient reported in <Table 2> between the two variables is 0.9791, implying that the behavior of *informed* investors, such as hedge funds, is closely related to the productivity in the U.S. business sector.

The correlation coefficient between *TFP* and business output (*Output*) is 0.9807, indicating that the difference in the various productivity measures is minor from an empirical perspective. The correlation coefficient between *HFI* and *TFP_Util* is also very high, at 0.9502. However, the correlation coefficient between *SNP500* and *TFP* is only 0.6598. Therefore, the stock market index seems to be much less related to the real economic productivity than the hedge fund index. If the stock market is somewhat informationally inefficient, then the stock prices reflect noisy information.

## 4. Empirical Results

In this section, I explain the results of the SVAR with a long-run restriction and the VECM without any restrictions as a robustness test. <Table 3> reports the estimation results of the VAR with a lag order of one, and <Table 4> summarizes the Granger Causality test between the two endogenous variables.

In the first column of <Table 3> predicting Δ*TFP*, the t-statistics of one-quarter lagged Δ*Ln*(*HFI*) is 3.9036, meaning that the differenced hedge fund index is statistically significant to predict the current changes in TFP productivity. Moreover, the hedge fund index is statistically stronger than the contemporaneous and lagged *SNP500* incorporating behaviors of all informed and uninformed investors. On the other hand, in the second column of <Table 3> predicting Δ*Ln*(*HFI*), the lagged Δ*TFP* is not statistically significant to predict the current return or performance of the hedge fund index. The Granger Causality tests in <Table 4> also present results similar to those in <Table 3>, where the first null hypothesis is rejected, but the second null hypothesis cannot be rejected. Therefore, I conclude that informed investors, such as hedge funds, would recognize news shocks on economic productivity in advance and move faster than other market participants.

<Table 5>, and <Figure 3> and <Figure 4> present the orthogonalized impulse responses with a long-run restriction, following Beaudry and Portier (2006), and the forecast error variance decompositions. A long-run restriction is imposed to identify the effects of a news shock on productivity. The left (right) panel of <Table 5> shows the results of the Structural (Cholesky) decompositions. I use Structural or Cholesky decomposition methods of the covariance matrix in error terms to make one underlying shock to be uncorrelated with the other shock.

First, the upper-right panels of <Figure 3> and <Figure 4> show the impulse responses of Δ*TFP* to a shock of Δ*Ln*(*HFI*) where the responses converge to zero regardless of imposing the long-run restriction in a year. That is, the effects of changes in hedge fund indexes on Δ*TFP* are significant^{7)} in the following quarter but almost disappear in 2 quarters after the shock of Δ*Ln*(*HFI*). Second, the lower-left panels present the effects of Δ*TFP* shock on the returns in hedge fund indexes, Δ*Ln*(*HFI*). They appear insignificant even in the first quarter after the shock and gradually disappear within a year.

The variance decompositions in <Table 5> also show similar results between two variables. In the first period, approximately 10.906% of the variation in Δ*TFP* is from shocks to the Δ*Ln*(*HFI*), and the contribution of Δ*Ln*(*HFI*)to the variation in Δ*TFP* converges to 18.380% when a Structural decomposition method is employed. In the case of Δ*Ln*(*HFI*), approximately 13.341% of the variation in Δ*Ln*(*HFI*) is from shocks to Δ*TFP* in the first period, and it converges to 13.904% and becomes stable after 3 quarters. Therefore, I conclude that hedge funds, as informed investors, would be able to know the news on future productivity in advance, so they move faster than any other agents in the economy.

<Table 6> presents the estimation results using the VECM with a lag order of one.^{8)}^{,} Beaudry and Portier (2006) identify news shocks inherent in the stock market index with the VECM to explain a factor causing future business cycle fluctuations. They argue that changes in technological opportunities are the main driver resulting in the business cycle fluctuations, and then stock market participants may notice these changes in advance. This study also employs the VECM as a robustness test because the result from a cointegration test with the endogenous variables in level is not statistically significant but the 1^{st} differenced variables are cointegrated from the test. Following the literature, current and lagged *UM_SENT* and *SNP500* are also included as exogenous variables.

In <Table 6>, the cointegrating vector (*β*) and the adjustment vector (*α*) are statistically significant, and represent the long-run equilibrium and short-run adjustment relationship between Δ(*TFP)* and Δ*Ln*(*HFI*), respectively. In the first equation with a dependent variable Δ(Δ*TFP*), the current and lagged stock market index, *SNP500* and *SNP500(-1)* are statistically significant although the lagged 2^{nd} differenced hedge fund index, Δ(Δ*Ln*(*HFI*)), is not statistically significant, implying that stock market participants may capture well the future economic conditions in advance, so information from the stock market seems to be useful to expect future productivity. In the second equation with a dependent variable Δ(Δ*Ln*(*HFI*)), the current and lagged stock market index, *SNP500* and *SNP500(-1)* are also statistically significant, implying that the changes in performance of hedge funds are related to the stock market performance.

<Table 7> and <Figure 5> show the results of the impulse responses and forecast error variance decompositions over the next 10 or 20 quarters in the VECM with a lag order of one. The upper-right panel of <Figure 5> presents the impulse response of Δ*TFP* to a shock of Δ*Ln*(*HFI*) where the responses converge to 1.1634 in a year. The magnitude of the impulse responses from 1^{st} quarter to 2^{nd} quarter is fairly large, at 1.5236, compared to the magnitudes of 0.8256 and 0.7932 from the SVAR model in <Table 5>. The lower-left panel of <Figure 5> shows the impulse response of Δ*Ln*(*HFI*) to a shock of Δ*TFP* where the responses converge to 0.1034 in a year. The forecast error variance decompositions in <Table 7> also indicate that approximately 65% of the variation in Δ*TFP* is from a shock to Δ*Ln*(*HFI*) in the horizon of 20 quarters, whereas approximately 18% of the variation in Δ*Ln*(*HFI*) is explained by a shock to the Δ*TFP*.

In summary, the findings from both SVAR and VECM models indicate that the trading behaviors of hedge funds are more likely to forecast the changes in TFP. In other words, information inferred from *informed* traders is more valuable for predicting future fundamentals or productivity than that inferred from the behaviors of all informed and uninformed traders. However, its predictive strength in the opposite direction appears to be limited.

## 5. Conclusions

This study examines the dynamic relationship between economic productivity, *TFP*, and hedge funds’ behaviors, *HFI*. The investment behaviors of hedge funds as *informed* traders quickly capture the news shock about future production growth; therefore, the information inferred from the hedge fund index returns is useful for predicting future economic productivity or economic conditions. Using a quarterly series of *TFP* and hedge fund index returns, this study finds that a predictive power exists in the behavior of hedge funds on real economic productivity. The findings from the SVAR and VECM models are robust regardless of whether a long-run restriction is imposed.

## References

## Notes

^{1)}

Examples of investment strategies by hedge funds include 1) global macro, 2) convertible arbitrage, 3) event-driven, 4) long/short equity, 5) emerging markets, 6) equity market neutral, 7) fixed income arbitrage, 8) multi-strategy, and so on (https://lab.credit-suisse.com/#/en/index/SECT/SECT/faq). For example, hedge funds may take equity market neutral positions in anticipation of a bear market, or actively bet on directional market moves. Therefore, hedge funds do not necessarily target certain types of firms that are highly affected by macro-level productivity compared to the firms included in the S&P500 index.

^{2)}

I include the real quarterly S&P500 composite stock market index as an exogenous variable to measure the marginal effect of investment behaviors by hedge funds, acting as informed traders, on TFP or productivity. This approach intentionally excludes the influence of mixed investment behaviors by all informed and uninformed traders. Of course, the S&P500 index could be included as an endogenous variable in the SVAR and VECM. However, this approach would make the system of equations more complex, leading to less clear interpretations. Additionally, to obtain comparable results, I adhere to the methodology and variable constructions suggested by Beaudry and Portier (2006), who use only two endogenous variables: the S&P500 index and TFP.

^{3)}

Source: The Barclay Hedge Fund database (https://www.barclayhedge.com/solutions/assets-under-management/); The reported aggregated AUM size does not include the AUM of funds of funds, which is approximately $335.2 billion as of 2nd quarter 2023.

^{4)}

The CS reports an aggregated composite index (i.e., The Credit Suisse AllHedge Index) and ten style-based sub-sector indexes since January 1994. In this study, I use the aggregated composite index which is an asset-weighted index set to 100 as of January 1994. The CSHF index is a rules-based measure of an investable portfolio, with its constituents rebalanced semi-annually according to the sector weights of the index.

^{5)}

However, no standardized definitions for the strategies are used.

^{6)}

The Credit Suisse Hedge Fund Indexes FAQ(https://lab.credit-suisse.com/#/en/index/HEDG/HEDG/faq).

^{7)}

The mean (median) of Δ*TFP* is 0.8121 (0.8915), and the standard deviation is 3.0643.

^{8)}

JMulTi software is used for VECM estimation (http://www.jmulti.de/).