The Relationship between Macroeconomic Volatility and Growth: Dynamics, Country Interactions and Nonlinearities

Transcription

1 The Relationship between Macroeconomic Volatility and Growth: Dynamics, Country Interactions and Nonlinearities STEVEN TRYPSTEEN University of Nottingham This paper investigates the relationship between macroeconomic volatility and growth for a set of OECD countries. It develops a time series model that accounts for dynamics in the volatility effect, country interactions, nonlinearities in the growth process, structural breaks and heterogeneous effects. The modelling process allows to distinguish between domestic and external volatility and its respective effect on growth. This is done by augmenting a country specific GARCH-M model of growth with lags of volatility, cross-country weighted averages, nonlinear terms and shift dummies. The results show that domestic volatility is positively associated with growth and that external volatility is negatively associated with growth. Keywords: GARCH-M, Global VAR, Structural breaks, bounce back effect. JEL: C32, C5, E32. The Appendix is available on Correspondence to: Sir Clive Granger building Room B45, School of Economics, The University of Nottingham, University Park, NG7 2RD, Nottingham, United Kingdom. e: steven.trypsteen@nottingham.ac.uk

2 1 Introduction The relationship between macroeconomic volatility and growth is a widely debated topic in the economics literature. Schumpeter (1934), for example, emphasized the idea of creative destruction leading to a positive relationship, whereas Arrow (1962) highlights the idea of learning-by-doing leading to a negative relationship. More recent theoretical work also covers the real option literature where firms view their investment choices as a series of options. As volatility increases the option value of delay, firms postpone their investment and so growth is negatively affected (Bernanke, 1983; Bloom, 2009). As the relationship is theoretically ambiguous, many researcher attempted to settle the question empirically. The seminal empirical paper by Ramey and Ramey (1995) investigates the relationship with panel growth regressions and find that volatility affects growth negatively, especially in OECD countries. Their measure of volatility is endogenously determined and constant over time. 1 Martin and Rogers (2000) also investigate the relationship with panel growth regressions and find a negative effect of volatility in developed countries. Their measure of volatility, in contrast to Ramey and Ramey (1995), is the unconditional standard deviation over a five year period. A drawback of panel growth regressions, however, is that macroeconomic volatility is assumed to be constant over an arbitrary time period. A number of studies acknowledge the time varying nature of volatility and use the Generalised Autoregressive Heteroscedasticity in Mean (GARCH-M) model developed by Engle, Lilien, and Robins (1987) to investigate the relationship between macroeconomic volatility and growth. The GARCH-M model consists of two equations, namely a conditional mean and a conditional variance equation. The conditional variance is endogenously determined and depends on the squared lag of the error term and a lag of the conditional variance. The model also allows for feedback from the conditional standard deviation, which is the time-varying measure of volatility, to growth. Note that this model can be estimated on a country-by-country basis and so it does not impose homogeneous effects as with panel growth regressions. Studies that apply the GARCH-M framework, however, arrive at conflicting conclusions. Some find that output volatility is positively associated with growth (Fang & Miller, 2014; Fountas, Karanasos, & Kim, 2006; Grier, Henry, Olekalns, & Shields, 2004; Lee, 2010), negatively associated with growth (Bredin & Fountas, 2009; Fang, Miller, & Lee, 2008; Henry & Olekalns, 2002) or find no statistically significant relationship (Fang et al., 2008; Grier & Perry, 2000). All these studies, however, ignore a number of issues that could be important in identifying the 1 In a later section of their paper, they do consider a time varying measure of volatility. The time variation, however, is only due to government spending. 1

3 true relationship between macroeconomic volatility and growth. First, volatility can affect growth, but it takes time to work through into growth. Existing studies using the GARCH-M framework only include contemporaneous or the one period lag of volatility in the model. A more realistic econometric model, however, should allow for more dynamics in the effect of volatility. Second, as output is strongly correlated across countries (Kose, Otrok, & Whiteman, 2008), ignoring these country linkages could lead to bias. 2 Moreover, not only domestic volatility could affect domestic growth, the volatility in other countries, i.e. external volatility, could also affect the domestic growth rate. Third, there is evidence that output recovers strongly following a recession, sometimes called the bounce-back effect (Beaudry & Koop, 1993). This nonlinearity in the growth process can lead to a spurious positive relationship between output volatility and growth and should therefore be incorporated in the econometric model. Moreover, volatility could be more affected by negative shocks than positive shocks. This potential nonlinearity in the variance of growth should also be addressed. 3 Fourth, structural breaks in the mean and variance of output growth are an important characteristic of post war output data (Nordhaus, 2004; Stock & Watson, 2005). Moreover, ignoring structural breaks biases the autoregressive parameters towards 1 (Perron, 1989). Ignoring structural breaks in the time-varying variance of a GARCH model has a similar effect (see Hillebrand, 2005; Lamoureux & Lastrapes, 1990). It is therefore important to account for this feature. 4 This paper investigates the relationship between output volatility and growth for 13 OECD countries 5 while simultaneously accounting for the four issues discussed above, namely dynamics in the effect of volatility, country linkages, nonlinearities in the growth process and structural breaks. To that end it applies the standard GARCH-M framework and augments it accordingly. A first modification is to include lags of volatility to allow for a dynamic effect of volatility on growth. Cross-country weighted averages of growth and volatility are included to account for cross-country interactions. As including cross-country weighted averages can be interpreted as allowing for unobserved common factors, it is a useful way to account for country interactions (Dées, Di Mauro, Pesaran, & Smith, 2007). This technique also allows to distinguish between 2 A study that takes country linkages into account is Lee (2010). This study applies a dynamic panel GARCH-M model for the G7 where contemporaneous feedback in the variance equation between countries is accounted for. This framework, however, does not account for dynamics in the volatility effect, nonlinearities, structural breaks and heterogeneous effects. 3 Henry and Olekalns (2002) estimate an asymmetric GARCH-M for the United States and also allow for the bounce-back effect. Their empirical model, however, does not allow for dynamics in the effect of volatility, country linkages and structural breaks. 4 Fang et al. (2008) estimate a GARCH-M model for the G7, except France, and include shift dummies in the mean and variance equation to account for structural breaks. This framework, however, does not take dynamics in the effect of volatility, country interactions and nonlinearities into account. 5 The countries considered are Canada (CAN), Finland (Fin), France (FRA), Germany (GER), Greece (GRE), Italy, (ITA), Japan (JAP), the Netherlands (NED), Norway (NOR), Portugal (POR), Sweden (SWE), the United Kingdom (UK) and the United States (US). 2

4 volatility that stems from internal shocks (domestic volatility) and volatility that originates from outside the country (external volatility). The bounce-back effect discussed above is accounted for by including a measure of the current depth of recession as in Beaudry and Koop (1993). To incorporate the asymmetric effect of positive and negative shocks to the variance of growth, the variance equation includes a nonlinear term as in Glosten, Jagannathan, and Runkle (1993). Finally, the model also includes shift dummies in the mean and variance equation to account for potential structural breaks. The number and timing of the breaks are country specific and determined before the estimation process based on Bai and Perron (2003). An additional benefit of the modelling approach, compared to panel growth regressions, is that it allows for heterogeneous effects across countries. As the augmented GARCH-M model is estimated on a country-by-country basis, the effect of volatility and the short-run dynamics are not imposed to be equal across countries. The main finding of the paper is that domestic volatility is positively associated with growth, whereas external volatility is negatively associated with growth. The effect of domestic volatility is positive in all countries and statistically significant at conventional levels in all countries, except Japan. The effect of external volatility has a negative coefficient in all countries and is statistically significant at conventional levels in 6 out of 13 countries, namely Canada, France, the Netherlands, Sweden, the United Kingdom and the United States. The results also indicate that country linkages are important as all estimated models include cross-country weighted averages of growth. The augmented GARCH-M model therefore captures rich dynamics with lagged own country growth and cross-country weighted averages of growth. The results related to the nonlinear terms included in the model show that nonlinearities in the mean and variance are not that important. There is no clear evidence of a bounce back effect and only in Germany and the United States there is evidence that negative shocks affect the variance more than positive shocks. The results also show that structural breaks in the mean and variance are important for many countries. Six out of 13 countries experience at least 1 break in the mean of growth and 9 out of 13 countries experience at least 1 break in the variance of growth. The timing, direction and size of the breaks differs across countries. One relatively common feature is that 8 countries experience a negative break during the 1980s, namely Canada, Finland, France, Italy, Portugal, Sweden, the United Kingdom and the United States. The rest of the paper is organised as follows. Section 2 describes the augmented GARCH-M model that is used to investigate the relationship between macroeconomic volatility and growth. It also derives the long-run effect of domestic and external output volatility. Section 3 discusses the data properties, model selection procedure and estimation results. Section 4 concludes. 3

5 2 Modelling the relationship between macroeconomic volatility and output growth The model To investigate the relationship between macroeconomic volatility and output growth, I use the standard GARCH-M model, but modify it in four important ways. First, I allow for volatility dynamics by including lagged volatility terms to the standard GARCH-M model as volatility is likely to affect growth with a lag. A standard GARCH-M model only allows for contemporaneous volatility to affect the dependent variable and so cannot incorporate the potential inertia effects of volatility on growth. Second, I include cross-country weighted averages of growth and volatility to account for country interactions. The co-movement of output across countries is a well-documented feature of post-war output data. Kose et al. (2008), for example, estimate common and country-specific components in output, investment and consumption of the G7 countries with a Bayesian dynamic latent factor model. They find that the common factor explains a lot of variation in these macroeconomic series. They also find that the common factor becomes more important over time and so the increased trade and financial links in recent decades lead to an increase in the co-movement of these series. As including cross-country weighted averages can be interpreted as allowing for unobserved common factors (Dées et al., 2007), it is a useful way to account for country interactions. This technique also allows to distinguish between volatility that stems from domestic shocks (domestic volatility) and volatility that originates from external forces (external volatility). Third, I allow for nonlinear effects in the mean and the variance of growth. An important form of nonlinearity for the relationship between output volatility and growth is the possibility of a bounce-back effect, i.e. output recovers strongly following a recent recession. Ignoring such an effect, could lead to finding a spurious positive relationship between output volatility and growth. Indeed, after a negative shock, growth increases faster leading to increased volatility and so a positive correlation between output volatility and growth. Beaudry and Koop (1993) develop an effective way to model the bounce back effect which is also applied here. They create a variable called current depth of recession (CDR) defined as cdr t = max{y t j } j 0 Y t, where Y t is output, and include lags of it as regressors. A positive effect of CDR terms implies that output growth is faster in the recovery phase of a downturn. The augmented GARCH-M model also includes a nonlinear term in the variance equation. In particular, I use the asymmetric GARCH specification as in Glosten et al. (1993) where negative 4

6 shocks can affect the variance in a different way then positive shocks. If this form of nonlinearity was ignored, then the measure of volatility would be biased. Fourth, each country model includes country-specific shift dummies. An important property of output growth is the occurrence of structural breaks in both the mean, as demonstrated by the literature on productivity slowdowns (see for example Nordhaus, 2004) and the variance, as demonstrated by the literature on the great moderation 6 (see for example Kim & Nelson, 1999; McConnell & Perez-Quiros, 2000; Stock & Watson, 2005). Moreover, accounting for structural breaks is also important for econometric reasons. Perron (1989) shows that ignoring structural breaks biases the autoregressive parameters towards 1. Ignoring structural breaks in the timevarying variance of a GARCH model has a similar effect (see Hillebrand, 2005; Lamoureux & Lastrapes, 1990). More formally, the model for country i = 1, 2,..., N, is given by l m y it = c i0 + c ik DM ikt + λ ik σ it k + σ 2 it = α i0 + k=0 p φ ik y it k + n k=0 λ ikσ it k s φ ik yit k + k=0 r τ ik cdr it k + ε it (2.1) f α ik DV ikt + η i ε 2 it 1 + θ i σit γ i ε 2 it 1I(ε it 1 < 0) (2.2) where y it is the growth rate for country i at time t, DM ikt is a possible shift dummy k in the conditional mean which is equal to 0 before the break date and equal to 1 on and after the break date, σ it k is the conditional standard deviation of growth in country i at time t k, which is the measure of domestic volatility, σ it k is the cross-country weighted average of the conditional standard deviation, which is the measure of external volatility, yit k is the cross-country weighted average of growth, cdr it k is the current depth of recession measure as in Beaudry and Koop (1993) and ε it is the error term where ε t (0, σt 2 ). The conditional variance of country i at time t, σit 2, depends on a constant α i0, possible shift dummies, DV ikt, which are equal to 0 before the break date and equal to 1 on and after the break date, the one period lag of the squared error term, the one period lag of the conditional variance and the asymmetric term where I(.) is the indicator function. 6 Bernanke (2004) argues that the economic literature has suggested three types of explanations for the reduction of output volatility i.e. the great moderation, namely (i) structural change, (ii) improved performance of macroeconomic policies and (iii) good luck, i.e. less shocks. 5

7 The cross-country weighted averages of growth and volatility are respectively defined as N yit = w ij y jt j=1 N σit = w ij σ jt j=1 N j=1 N j=1 w ij = 1 and w ii = 0 (2.3) w ij = 1 and w ii = 0 (2.4) where w ij is the weight given to country j in the weighted averages of country i. The weights sum to one and ones own growth or volatility is given a weight of 0. 7 Assuming that the cross-country weighted averages of growth and volatility account for all the country interactions, it follows that the error term, ε it, of each country is idiosyncratic and so the error term across countries is uncorrelated, i.e. COV [ε it, ε jt ] = E[ε it ε jt ] = 0 for i j. This also implies that the conditional standard deviation of country i, σ it, can be interpreted as domestic volatility as it only depends on country-specific terms. Furthermore, the cross-country weighted average of the conditional standard deviation of country i, σit, can then be interpreted as external volatility as it is a weighted averages of the domestic volatilities of other countries. Measuring the long-run effect of domestic and external volatility As it takes time for volatility to affect the growth rate, via the short run dynamics and the country linkages, it is important to consider the long-run effects of domestic and external volatility. In order to find the long-run effect of domestic and external volatility, I first stack the model, i.e. l m n y t = c 0 + c k DM kt + λ k σ t k + λ kσt k σ 2 t = α 0 + k=0 p φ k y t k + k=0 s φ k yt k + k=0 r τ k cdr t k + ε t (2.5) f α k DV kt + ηε 2 t 1 + θσt γε 2 t 1I(ε t 1 < 0) (2.6) y t = W y t (2.7) σ t = W σ t (2.8) where y t = ( y 1t,..., y Nt ) and similarly for other emboldened vectors. All the coefficient matrices are N N matrices, except the constants c 0 and α 0 which are N 1 vectors. The diagonal elements of these matrices are equal to the country-specific estimates and the off-diagonal 7 Note that the weights are assumed to be constant over time. In practice this is not a problem. I have chosen to use weights based on trade linkages between countries. It turns out that the relative amount of trade between countries is stable over the sample period. See the Appendix for details. 6

8 elements are equal to 0. The country specific weights, w ij, are collected in the weight matrix W. Substituting (2.7) in (2.5) gives l m n y t =c 0 + c k DM kt + λ k σ t k + λ kσt k (2.9) + p φ k y t k + s φ kw y t k + k=0 and by using the lag operator L the model in (2.9) can be written as r τ k cdr t k + ε t ( ) I N φ(l) φ (L)W y t = c 0 + y t = l c k DM kt + λ(l)σ t + λ (L)σt + τ (L)cdr t + ε t ( ) 1 ( I N φ(l) φ (L)W c 0 + l ) c k DM kt + λ(l)σ t + λ (L)σt + τ (L)cdr t + ε t (2.10) where φ(l) = φ 1 L + φ 2 L φ p L p, φ (L) = φ 0 + φ 1L + φ 2L φ sl s, λ(l) = λ 0 + λ 1 L λ m L m, λ (L) = λ 0 + λ 1L λ nl n and τ (L) = τ 1 L + τ 2 L τ r L r. By setting the lag operator to 0, we can find the short run effect or impact effect of domestic and external volatility for all countries, ( yt σ t )SR = λ 0 and ( yt ) σt SR = λ 0 (2.11) The long-run effect of domestic and external volatility is found by setting the lag operator to 1, ( yt σ t )LR ( yt ) σ t LR = = ( I N Φ(1) Φ (1)W ) 1λ(1) λlr (2.12) ( I N Φ(1) Φ (1)W ) 1λ (1) λ LR (2.13) As the model takes country interactions into account through the weight matrix W, the off-diagonal elements of the matrices with the long-run effects are nonzero. The diagonal elements are the longrun effects of country i s volatility on the growth rate of country i. The off-diagonal elements are the long-run effects of country j s volatility on the growth rate of country i. It follows that for each country there are N long-run effects. To arrive at the total long-run effect, i.e. taking the effect of each individual country into account, the row elements of λ LR and λ LR are summed. Thus the total long-run effects of domestic and external volatility on the individual countries output 7

9 growth are equal to λ T LR λ LR S and λ T LR λ LRS (2.14) where S is a N 1 vector of ones. 3 Estimating the relationship between macroeconomic volatility and output growth for OECD economies This section discusses the estimation results of the model in (2.1) (2.4) for 13 OECD countries and evaluates the total long-run effects in (2.14). Before presenting the actual results, however, it first gives an overview of the data and describes the model selection procedure. Properties of output growth As a measure of output, I use the seasonal adjusted monthly index of industrial production over the period February 1962 March 2015 from the OECD s Main Economic Indicators (MEI) database. Output growth is calculated as the monthly difference of the natural log of the index of industrial production. The 13 OECD countries considered are Canada (CAN), Finland (Fin), France (FRA), Germany (GER), Greece (GRE), Italy, (ITA), Japan (JAP), the Netherlands (NED), Norway (NOR), Portugal (POR), Sweden (SWE), the United Kingdom (UK) and the United States (US). Table 1 presents the summary statistics of the monthly growth rates for the 13 OECD countries over this period. 8 The table shows that Japan grew at the highest mean rate of 0.28% per month and the UK grew at the lowest rate of 0.09% per month. In terms of unconditional standard deviation of growth, Norway is the most volatile country, whereas the United States is the least volatile country. The monthly growth rates of industrial production for these OECD countries are also found to be stationary, serially correlated, conditional heteroscedastic and not normally distributed. 9 The cross-country weighted averages of growth and volatility, defined in (2.3) and (2.4) respectively, are calculated with the average share of total trade of country i with country j as weight, w ij. 10 Total trade is defined as the sum of exports and imports between country i and the other 8 For some countries there were outliers. The outliers are replaced with the values obtained through log-linear interpolation of the index of industrial production. See the Appendix for more details. 9 See the appendix for the results. 10 See the Appendix for the resultant cross-country trade weighted averages of growth plotted over time. This type of graphs cannot yet be produced for the cross-country trade weighted average of volatility as this is only possible after estimation. A weight matrix of all the trade weights used for the calculation can also be found in the Appendix. 8

10 Table 1: Summary statistics of the annualized monthly growth rates of industrial production (in %), 1962: :03 Obs. Median Mean Sta. Dev. Min Max CAN FIN FRA GER GRE ITA JAP NED NOR POR SWE UK US countries considered. Note that the trade weights are averaged over time and so constant. This assumption is justified as the trade weights are stable over time. 11 The data used to calculate the trade weights is import and export data from the IMF s Direction of Trade Statistics. Another important property of output growth series is the possibility of structural breaks, especially when investigating long time series. The procedure to determine breaks in the mean and the variance of output growth applied here is based on Bai and Perron (2003, p.15-16). To find breaks in the mean of a series x t, they recommend to first regress x t on a constant and accounting for potential serial correlation via non-parametric adjustment. 12 Based on this regression, the UDmax and W Dmax test statistics are used to see if at least one break is present. If the UDmax test statistic, the W Dmax test statistic or both are larger than the relevant critical value, then we can reject the null hypothesis of 0 breaks in favour of at least one break. If there is evidence of at least one break, then the SupF (l+1 l) statistic, where we test the null hypothesis of l+1 breaks given l breaks, is used sequentially to determine higher order breaks where I allow the maximum number of possible breaks to be The approach to find breaks in the variance of a series x t, following Herrera and Pesavento (2005) and Fang and Miller (2014), is to regress x t on a constant and on the mean break dummy or dummies found previously and to save the residuals, say ˆµ it. The UDmax, W Dmax and SupF (l+1 l) statistics are then used in the same way as discussed above based on the regression of the absolute values of ˆµ it on a constant. In order to account for country interactions in this procedure, I do not apply the above procedure directly on the growth rates, but on the growth 11 See the Appendix for plots of the trade weights over time. 12 The non-parametric adjustment is to estimate the model using a quadratic spectral kernel based HAC covariance estimation using prewhitened residuals. The kernel bandwidth is determined automatically using the Andrews AR(1) method. 13 The procedure allows for serial correlation and different variances of the residuals across regimes. Also, the trimming throughout the procedure is As the sample period is 1962: :03 (638 observation), each segment is about 8 years (96 observations) at a minimum. 9

11 rates where the effect of country interactions are taken out. This is done by regressing the output growth rate of country i on a constant and contemporaneous and lagged cross-country weighted averages of growth, where I allow for up to 6 lags and the optimal lag length is determined by the AIC. Table 2 shows the results of the break detection procedure, where panel A and B show the results for the mean and variance of output growth, respectively. The emboldened figures are statistics that are larger than their respective critical value. For the breaks in the mean of output growth, the results indicate that 4 countries experienced one break, 2 countries experienced 2 breaks and 7 countries experienced no break. There is not a real pattern in the timing of the breaks. The results for the breaks in the variance of output growth show that for four countries there is no break, for six countries there is one break, for two countries there are two breaks and for one country there are three breaks. Here too, there is not a clear pattern, except that many countries have a break in the 1980s. By creating shift dummy variables based on these results and including them in the regression model, we will be able to estimate the direction and size of the respective breaks. Model selection procedure To find the most parsimonious representation of the data, I follow a two step model selection procedure. The first step is to estimate the model in (2.1) (2.4) for various lag lengths of the different components of the model while setting the λ ik s to 0. The most general model considered is an asymmetric GARCH-M model with the t distribution as error distribution 14 with six autoregressive lags, a contemporaneous and six lags of cross-country weighted averages of growth and three cdr lags in the mean equation. Note that domestic volatility is always included up to three lags. The mean and variance equation of all these models are jointly estimated 15 with maximum likelihood using the Marquardt optimization algorithm. As with any iterative estimation method, however, the algorithm could stop at a local instead of the global maximum. To counter this problem, all the models are estimated with various initial values and I choose the initial values that generates the largest log likelihood. 16 Information Criteria (AIC) is then picked. The well-specified model that minimises the Akaike A model is well-specified if the mean and variance of the standardized residuals are equal to 0 and 1, respectively, the distribution of the residuals 14 The model with the t-distribution as error distribution has one more parameter compared to the model where the error distribution is assumed to be normal, i.e. the number of degrees of freedom, and so is more general. If these are estimated to be large, then the t-distribution collapses to the normal distribution. 15 As the Hessian of the log-likelihood function of the GARCH-M model is not block diagonal, the mean and variance parameters are correlated and so all the parameters need to be estimated simultaneously. 16 I estimate the models with the OLS estimates of the mean equation as initial values for the mean parameters and with 50% of these OLS estimates as initial values. 10

12 Table 2: Results of the break detection procedure Panel A: Mean of output growth CAN FIN FRA GER GRE ITA JAP 5% critical value UDmax W Dmax SupF (2 1) SupF (3 2) SupF (4 3) SupF (5 4) Break M M01 Break M06 NED NOR POR SWE UK US 5% critical value UDmax W Dmax SupF (2 1) SupF (3 2) SupF (4 3) SupF (5 4) Break M M M M04 Break M06 Panel B: Variance of output growth CAN FIN FRA GER GRE ITA JAP 5% critical value UDmax W Dmax SupF (2 1) SupF (3 2) SupF (4 3) SupF (5 4) Break M M M M12 Break M04 NED NOR POR SWE UK US 5% critical value UDmax W Dmax SupF (2 1) SupF (3 2) SupF (4 3) SupF (5 4) Break M M M M M02 Break M M03 Break M08 Note: - Emboldend statistics are statistics that are larger than their respective critical value at the 5% significane level. 11

13 Table 3: Exogeneity test of y it CAN FIN FRA GER GRE ITA JAP ζ ( ) ( ) ( ) ( ) ( ) ( ) ( ) NED NOR POR SWE UK US ζ ( ) ( ) ( ) ( ) ( ) ( ) Note: - t-statistics based on White heteroskedasticity-consistent standard errors are in brackets. corresponds to the one assumed in the estimation procedure and there is no evidence of serial correlation and conditional heteroscedasticity in the standardized residuals. 17 Note that the contemporaneous y it is included as a potential regressor. If this variable, however, cannot be treated as exogenous, then we should not include it. In order to test for this I apply the standard Hausman (1978) test on the model in (2.1)-(2.4) where the GARCH-M framework is excluded. 18 The test involves two steps. First, I estimate the reduced form of y it and save the residuals, say ˆν it. The reduced form of y it includes all the variables as in the original mean model plus an extra autoregressive and cross-country weighted average lag. Then, I add the residuals ˆν it to the adjusted version of (2.1)-(2.4) and test if the coefficient on these residuals, denoted as ζ, is significant. If they are not, then this implies that y it is exogenous. Table 3 presents the estimated ζ and corresponding standard errors. For all OECD countries, the null hypothesis that the estimated coefficient ζ is 0 cannot be rejected at the 5% level, except for Canada and the Netherlands. As Canada and the Netherlands are relatively small in terms of GDP, I ignore this finding and treat yit as exogenous for all countries. The second step of the model selection procedure is to estimate the complete model in (2.1) (2.4) with the lag structure and nonlinear structure found in step 1 and using the estimates found in step 1 as initial values. As the initial values for the λ ik s, I use 1, 0.99, 0.98,..., 0.98, 0.99, 1 and pick the initial value that maximises the log likelihood. Estimation results Table 4 shows the estimation results of the model of (2.1) (2.4) for each country where panel A and B show the estimates for the mean and variance equation, respectively I also check if the Zero-Information-Limit Condition (ZILC) highlighted by Nelson and Startz (2007) and Ma, Nelson, and Startz (2007) does not hold for the estimated GARCH(1,1) models. See the appendix for more details. 18 Hausman (1978) proposed to test if OLS and 2SLS estimates differ significantly. If they do not, then the variable can be regarded as exogenous. As GARCH-M models cannot be estimated with OLS nor 2SLS, I need to exclude the GARCH-M framework in (2.1)-(2.4) in order to apply the test. 19 The results of the residual diagnostic tests can be found in the Appendix. 12

14 The coefficients of interest are the effects of domestic volatility, λ k, and the effects of external volatility, λ k. The individual lags of domestic and external volatility, however, are unlikely to be individually statistically significant due to multicollinearity. The results show that this is indeed the case. The long run propensity of both volatility s, given in (2.3) and (2.4), are therefore more informative and these are shown in Table 3. The standard errors of the long-run effects are calculated using the delta method. It turns out that the effect of domestic volatility, λ T LR, is positive in all countries and statistically significant at conventional levels in all countries, except Japan. The effects range from in Canada to in Greece. The effect of external volatility has a negative coefficient in all countries and is statistically significant at conventional levels in 6 out of 13 countries, namely Canada, France, the Netherlands, Sweden, the United Kingdom and the United States. The effects range from in Japan to in Greece. 20 I now return to the individual country models of Table 4 and discuss the individual components of the model. Panel A shows that the cross-country weighted averages of growth show up significantly across all the equations and so country linkages are important. This implies that the model captures sophisticated dynamics with lagged own country growth and cross-country weighted averages of growth. The coefficients on the included mean shift dummies all show up statistically significant, except the second break for Japan. Greece, Japan, the Netherlands, Norway and Portugal all experienced a negative break in the mean growth rate. The United states experienced a positive break at the start of the 1990s and a negative one at the end of the 1990s. There is, however, no pattern in the timing or size of the breaks in the mean growth rate across countries. CDR terms enter the model for 6 out of 13 countries, namely Finland, France, Greece, Italy, Norway and the United States. 21 Even thought CDR terms are included for these countries, there is not much evidence for a bounce back effect. The United States and Finland have each one lag of the CDR term in their model. For the United states the coefficient on CDR t 1 is positive, but not statistically significant. For Finland the coefficient is statistically significant, but has a negative sign. For the countries with more than 1 CDR lag, the CDR lags seem to cancel each other out. This shows that the bounce back effect is not important here. The estimates of the variance equation in Panel B of Table 4 show that the variance model is the ARCH(1) model for all countries, except for Italy and Japan where the variance model is a GARCH(1,1) model. The nonlinear term enters the model for Finland, Germany, Italy and the United States. It is always positive, but it is only statistically significant in Germany and the United States. Thus we can conclude that only for these two countries negative shocks increase 20 Graphs of the estimated measure of domestic and external volatility over time can be found in the Appendix. 21 See the Appendix for the calculated CDR series over time. 13

15 volatility more than negative shocks. Also, all models are estimated with the t-distribution as error distribution, except for Canada, the Netherlands and Norway. All the variance shift dummies included in the models are statistically significant at the 5% level. Each country, however, has its own particular break pattern. One thing that stands out, though, is that 8 of the 13 OECD countries experience a negative break in during the 1980s, namely Canada, Finland, France, Italy, Portugal, Sweden, the United Kingdom and the United States. 14

16 Table 4: Estimation results Panel A.1: Mean Equation CAN FIN FRA GER GRE ITA JAP c ( ) ( ) ( ) ( ) ( ) ( ) ( ) c ( ) ( ) [1979M06] [1971M01] c ( ) [1991M06] λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) φ ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) φ ( ) ( ) τ ( ) ( ) ( ) ( ) τ ( ) ( ) ( ) τ ( ) ( ) ( ) 15

17 Table 3: Continued Panel A.2: Mean Equation NED NOR POR SWE UK US c ( ) ( ) ( ) ( ) ( ) ( ) c ( ) ( ) ( ) ( ) [1974M09] [1997M11] [1990M09] [1991M04] c ( ) [1999M06] λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) φ ( ) ( ) φ ( ) φ ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) φ ( ) ( ) ( ) ( ) φ ( ) ( ) τ ( ) ( ) τ ( ) 16

18 Table 2: Continued Panel B: Variance Equation CAN FIN FRA GER GRE ITA JAP α ( ) ( ) ( ) ( ) ( ) ( ) ( ) α ( ) ( ) ( ) ( ) [1984M11] [1991M08] [1973M04] [1985M12] α ( ) [1981M04] η ( ) ( ) ( ) ( ) ( ) ( ) ( ) θ ( ) ( ) γ ( ) ( ) ( ) ν ( ) ( ) ( ) ( ) ( ) ( ) NED NOR POR SWE UK US α ( ) ( ) ( ) ( ) ( ) ( ) α ( ) ( ) ( ) ( ) ( ) [1979M04] [1982M09] [1970M09] [1972M01] [1984M02] α ( ) ( ) [1989M11] [1987M03] α ( ) [1999M08] η ( ) ( ) ( ) ( ) ( ) ( ) γ ( ) ν ( ) ( ) ( ) ( ) Notes: - The model is: l m n y it = c i0 + c ik DM ikt + λ ik σ it k + λ ik σ it k + k=0 k=0 p s r φ ik y it k + φ ik y it k + τ ik cdr it k + ε it k=0 f σit 2 = α i0 + α ik DV ikt + η i ε 2 it 1 + θ iσit γ iε 2 it 1 I(ε2 it 1 < 0) N yit = w ij y jt j=1 N σit = w ij σ jt j=1 N w ij = 1 and w ii = 0 where y it is the growth rate for country i at time t, DM ikt and DV ikt is a possible shift dummy k in the conditional mean and variance, respectively, y it is the growth rate for country i at time t, σ it k is the conditional standard deviation of growth in country i at time t k, which is the measure of domestic volatility, σit k is the cross-country weighted average of the conditional standard deviation, which is the measure of external volatility, yit k is the cross-country weighted average of growth, cdr it k is the current depth of recession measure as in Beaudry and Koop (1993), ε it is the error term, w ij is the average share of total trade of country i with country j and I(.) is the indicator function. j=1 - Standard errors are in brackets and the break dates are in square brackets., and denote statistical significance at the 10%, 5% and 1% level, respectively. ν is the parameter for the degrees of freedom. 17

19 Table 3: The total long-run effect of domestic and external volatility CAN FIN FRA GER GRE ITA JAP λ T LR ( ) ( ) ( ) ( ) ( ) ( ) ( ) λ T LR ( ) ( ) ( ) ( ) ( ) ( ) ( ) NED NOR POR SWE UK US λ T LR ( ) ( ) ( ) ( ) ( ) ( ) λ T LR ( ) ( ) ( ) ( ) ( ) ( ) Notes: - The standard errors are in brackets and are obtained with the delta method., and denote statistical significance at the 10%, 5% and 1% level, respectively. 4 Conclusion As the relationship between output volatility and growth is theoretically ambiguous empirical analysis can help to discriminate between various theories. This paper investigates the relationship for 13 OECD countries by employing the GARCH-M framework, but adjusting the framework so that it accounts for important issues ignored by existing empirical studies. First, the GARCH-M model is augmented with lags of volatility to account for dynamics in the effect of volatility. Second, the model includes cross-country trade weighted averages of growth and volatility to account for cross-country interactions. This approach also allows to distinguish between domestic and external volatility and its respective effect on growth. Third, by including a measure of the current depth of recession the model accounts for the bounce back effect. The variance equation also includes a nonlinear term that allows negative shocks to have a different effect on the variance than positive shocks. Fourth, the mean and variance equation of the augmented GARCH-M model include shift dummies to account for structural breaks. The analysis has a number of interesting findings. First, domestic volatility is positively associated with growth and the effect is statistically significant in all countries. External volatility, in contrast, is negatively associated with growth and the effect is statistically significant in 7 out of the 13 OECD countries. Second, the analysis shows that cross-country linkages are important in a model of output growth. Third, there is not much evidence for a bounce back effect. Asymmetric effects in the variance also play a minor role. Finally, structural breaks in the mean and variance of growth are important in many countries. The timing, size and direction of the breaks is country specific. Once exception is that 8 out of 13 OECD countries experienced a negative break in the variance in the 1980s. 18

20 References Arrow, K. J. (1962). The Economic-implications of Learning By Doing. Review of Economic Studies, 29 (80), Bai, J. S., & Perron, P. (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18 (1), Beaudry, P., & Koop, G. (1993). Do Recession Permanently Change Output. Journal of Monetary Economics, 31 (2), Bernanke, B. S. (1983). Irreversibility, Uncertainty, and Cyclical Investment. Quarterly Journal of Economics, 98 (1), Bernanke, B. S. (2004). The great moderation. At the meetings of the Eastern Economic Association, Washington, DC February 20, Bloom, N. (2009). The Impact of Uncertainty Shocks. Econometrica, 77 (3), Bredin, D., & Fountas, S. (2009). Macroeconomic uncertainty and performance in the European Union. Journal of International Money and Finance, 28 (6), Dées, S., Di Mauro, F., Pesaran, M. H., & Smith, L. V. (2007). Exploring the international linkages of the euro area: A global var analysis. Journal of Applied Econometrics, 22 (1), Engle, R. F., Lilien, D. M., & Robins, R. P. (1987). Estimating Time-varying Risk Premia In the Term Structure - the Arch-m Model. Econometrica, 55 (2), Fang, W. S., & Miller, S. M. (2014). Output Growth and its Volatility: The Gold Standard through the Great Moderation. Southern Economic Journal, 80 (3), Fang, W. S., Miller, S. M., & Lee, C. (2008). Cross-country evidence on output growth volatility: Nonstationary variance and GARCH models. Scottish Journal of Political Economy, 55 (4), Fountas, S., Karanasos, M., & Kim, J. (2006). Inflation uncertainty, output growth uncertainty and macroeconomic performance. Oxford Bulletin of Economics and Statistics, 68 (3), Glosten, L., Jagannathan, R., & Runkle, D. (1993). On the Relationship between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. Journal of Finance, 48 (5), Grier, K. B., Henry, O. T., Olekalns, N., & Shields, K. (2004). The asymmetric effects of uncertainty on inflation and output growth. Journal of Applied Econometrics, 19 (5), Grier, K. B., & Perry, M. J. (2000). The effects of real and nominal uncertainty on inflation and output growth: Some GARCH-M evidence. Journal of Applied Econometrics, 15 (1), Hausman, J. (1978). Specification tests in econometrics. Econometrica, 46 (6), Henry, O. T., & Olekalns, N. (2002). The effect of recessions on the relationship between output variability and growth. Southern Economic Journal, 68 (3), Herrera, A. M., & Pesavento, E. (2005). The decline in US output volatility: Structural changes and inventory investment. Journal of Business & Economic Statistics, 23 (4), Hillebrand, E. (2005). Neglecting parameter changes in GARCH models. Journal of Econometrics, 129 (1-2), Kim, C., & Nelson, C. (1999). Has the US economy become more stable? A Bayesian approach based on a Markov-switching model of the business cycle. Review of Economics and Statistics, 81 (4), Kose, M. A., Otrok, C., & Whiteman, C. H. (2008). Understanding the evolution of world business cycles. Journal of International Economics, 75 (1),