The Swiss franc safety premium

This paper applies a recent method proposed by Maggiori (The U.S. Dollar Safety Premium, 2013) to estimate the Swiss franc safety premium. The results show that the three-step instrumental variable approach as used by Maggiori does not work for the Swiss franc exchange rates. The price of risk estimates take unrealistic, negative values. One possible explanation is that the approach as it is used by Maggiori suffers from a measurement error for the expected exchange rate which represents a potential source of imprecision. By using the prediction of an augmented Fama regression to measure the expected exchange rate change, this measurement error can be avoided and the safety premium estimates become more realistic and closer to those obtained with a maximum likelihood-estimated GARCH approach. Overall, however, the GARCH approach still seems to be preferable to the instrumental variable approach. Electronic supplementary material The online version of this article (10.1186/s41937-017-0014-7) contains supplementary material, which is available to authorized users.

Notes: Correlation between monthly local stock market returns and exchange rate returns (end of period values) for the time period January 1990 to August 2011, depending on whether stock returns are below or above 0. Local stock market returns are calculated from the S&P 500 Index in case of the USD exchange rate index and the SPI for the two CHF exchange rates. ***, **, and * denote significance levels of 1, 5, and 10%, respectively, based on a t-test. A.3

C Robustness
For completeness, I provide here the results for the subperiods for (1) when the exchange rate change is included in the set of instruments and (2)

C.1 Set of Instruments Including the Exchange Rate
Here, I exclude the lagged exchange rate change as regressor in the zero-stage regression (and hence use the predictions of the original Fama model to construct the dependent variable in the second stage regressions). In return, I include the lagged exchange rate change in the set of instruments for the conditional covariance.
A.4 Notes: The estimates of the conditional covariance correspond to the fitted value of the first stage regression: Cov t r ω t+1 , e t+1 =α Z Z t (see equation (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, the lagged exchange rate change, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The zero stage regressions are estimated for each subsample separately. The first subsample (January 1990to December 1998 consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.
A.5 Notes: This table reports the results of the first stage regression, which regresses the ex-post covariance obtained from the zero stage regressions on a set of instruments (see equation (8)). This set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, the lagged exchange rate change, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The parameters are estimated by OLS using Newey-West standard errors with maximum lag order set equal to T 1/2 . The F-statistic and the Wald χ 2 test (plus the p-value for the Wald χ 2 test) are reported for the null hypothesis that all coefficients, except the constant, are jointly zero. The model is estimated separately for each subsample. The first subsample (January 1990 to December 1998) consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively.    (13)). The dependent variable is the CHF safety premium, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression estimated with the complete set of instruments. This consists of a constant, the dividend-price ratio, the lagged equity return, the lagged exchange rate change, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The model is estimated for each subsample separately. The first subsample (January 1990to December 1998 consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively.

C.2 Optimal Set of Instruments
Here, the instruments for each model are chosen such as to maximize the F-statistic in the first stage regression. Notes: This table reports the results of the first stage regression, which regresses the ex-post covariance obtained from the zero stage regressions on an "optimal" set of instruments (see equation (8)).
Instruments are selected such as to maximize the F-statistic. The set of possible instruments Z t consists of a constant, the dividendprice ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The parameters are estimated by OLS using Newey-West standard errors with maximum lag order set equal to T 1/2 . The F-statistic and the Wald χ 2 test (plus the p-value for the Wald χ 2 test) are reported for the null hypothesis that all coefficients, except the constant, are jointly zero. The model is estimated separately for each subsample. The first subsample (January 1990 to December 1998) consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively.
A.8   (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on the "optimal" set of instruments.
The possible set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, the lagged exchange rate change, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The zero stage regressions are estimated for each subsample separately. The first subsample (January 1990 to December 1998) consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.    (13)). The dependent variable is the CHF safety premium, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression estimated with the "optimal" set of instruments. The possible set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The model is estimated for each subsample separately. The first subsample (January 1990 to December 1998) consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively. A.10

C.3 Global Stock Market Index
This section reports the first and second stage results when the MSCI World index instead of the local stock market indices is taken to measure the investor's benchmark return.
Unlike in the benchmark case with the local stock market indices, one has now to be aware of the fact that the covariance between the exchange rate and the global stock market index converted into the respective currency incorporates direct exchange rate effects. For illustration, think of a situation where the local currency appreciates, while the global stock market index remains stable. In that case, the covariance between the exchange rate and the global stock market index converted into the local currency is positive even tough the value of the global stock market index has not changed. In the case of a safe currency, this implies that the covariance will tend to be overestimated.
While the price of risk estimates now take less extreme negative values when using the actual ex-post exchange rate change to measure the expected exchange rate change, the results in Tables A.8 and A.9 still support the main findings. With the use of the alternative measures for the expected exchange rate change, the standard deviations of the price of risk coefficients become smaller. When the expected exchange rate change is proxied by the zero stage regression, the results using global stock market returns get better (relative to the results using local stock market returns) for the full sample, while they get worse for the first subsample and are relatively stable for the second subsample. When considering the full sample and the second subsample, in three out of four cases the price of risk estimate for the CHF exchange rates lies between 2 and 3, i.e. values that are close to the ones found in the benchmark case. Overall, this robustness test supports the above finding that proxying the expected exchange rate change by the prediction of the zero stage regression yields more realistic and reliable estimations of the price of risk as compared to measuring the expected exchange rate change by the actual ex-post exchange rate. A.11

EUR/CHF
Notes: The estimates of the conditional covariance correspond to the fitted value of the first stage regression: Cov t r ω t+1 , e t+1 =α Z Z t (see equation (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.
A.12  (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.  Notes: This table reports the results of the second stage regression for the case of no time variation in the price of risk (see equation (13)). The dependent variables are the USD and CHF safety premium, respectively, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to -statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The number of observations is 259 for the full sample. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively.    (13)). The dependent variables are the USD and CHF safety premium, respectively, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The model is estimated for each subsample separately. The first subsample (January 1990 to December 1998) consists of 107 observations and the second subsample (January 1999 to August 2011) consists of 152 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively. A.15

C.4 Exclusion of Global Financial Crisis and Great Recession
This   (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.    (13)). The dependent variables are the USD and CHF safety premium, respectively, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The model is estimated for each subsample separately. The shortened full sample (January 1990 to July 2007) consists of 210 observations and the second subsample (January 1999 to July 2007) consists of 103 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively. A.18

C.5 Market Capitalization Weighted CHF Index
As an extension, this section provides the first and second stage results for a financially weighted CHF index, namely an MSCI market capitalization-weighted exchange rate and interest rate index based on the same weights as the USD indices. Overall, the results are similar to the ones of the trade-weighted CHF exchange rate. The slightly different pattern in the predicted conditional covariance (see Figure A.7) can be explained by the much higher weight that is now attributed to the USD, which is itself considered to be a safe currency.
Concerning the price of risk estimates, the findings are the same as for the other CHF exchange rates (see Table A  Notes: The estimates of the conditional covariance correspond to the fitted value of the first stage regression: Cov t r ω t+1 , e t+1 =α Z Z t (see equation (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.   (13)). The dependent variables are the USD and CHF safety premium, respectively, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The number of observations is 259 for the full sample, 107 for the first and 152 for the second subsample. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively. A.21

C.6 Sample Beginning in 1975
By complementing the interbank rates data with euro currency deposit rates data (Datastream) and replacing the local stock market indices by the MSCI World index (converted into the respective currency), the sample can be extended to go back to January 1975 (which corresponds to a total of 439 observations). This section reports the according first and second stage results.
Unlike in the benchmark case with the local stock market indices, one has now to be aware of the fact that the covariance between the exchange rate and the global stock market index converted into the respective currency incorporates direct exchange rate effects. For illustration, think of a situation where the local currency appreciates, while the global stock market index remains stable. In that case, the covariance between the exchange rate and the global stock market index converted into the local currency is positive even tough the value of the global stock market index has not changed. In the case of a safe currency, this implies that the covariance will tend to be overestimated. The predicted conditional covariance estimates plotted in Figure A.8 are indeed higher than the ones in the benchmark case reported in the main body of the paper (see Figure 5).
The price of risk estimates, on the other hand, are lower than the findings for the two subperiods in the benchmark case (see Table 7). Overall, however, the findings are the same as for the other CHF exchange rates: Proxying the expected exchange rate change by the prediction of the zero stage regression yields at least as or more realistic and reliable estimations of the price of risk as compared to measuring the expected exchange rate change by the actual ex-post exchange rate.
A.22 . 003 1975m1 1980m1 1985m1 1990m1 1995m1 2000m1 2005m1 2010m1 EUR/CHF Notes: The estimates of the conditional covariance correspond to the fitted value of the first stage regression: Cov t r ω t+1 , e t+1 =α Z Z t (see equation (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.  Notes: This table reports the results of the second stage regression for the case of no time variation in the price of risk (see equation (13)). The dependent variables are the USD and CHF safety premium, respectively, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The number of observations is 439 (January 1975 to August 2011). The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively. A.24

C.7 Sample Beginning in 1987
Historical data on the Swiss Performance Index (SPI) goes back to 1987. At the cost of a lower number of countries that can be included in the interest rate index, the beginning of the sample can be shifted from 1990 to 1987 (which corresponds to a total of 280 observations). This section reports the according first and second stage results. They hardly differ from the benchmark results in the main body of the paper.
A.25  (9)). In this first stage regression, the ex-post covariance obtained from the zero stage regressions is regressed on a set of instruments. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The two thin lines represent the 95% confidence band and are based on a two sided t-statistic with Newey-West estimates of the standard errors.
A.26   Notes: This table reports the results of the second stage regression for the case of no time variation in the price of risk (see equation (13)). The dependent variables are the USD and CHF safety premium, respectively, defined as the expected excess return of investing in the foreign risk-free asset by shorting the home risk-free asset. The expected exchange rate change used to calculate this expected excess return is proxied first by the actual exchange rate change, then by zero, and finally by the fitted value of the zero stage regression. The regressors are a constant and the estimate of the conditional covariance between stock returns and exchange rate changes from the first stage regression. The set of instruments Z t consists of a constant, the dividend-price ratio, the lagged equity return, plus a measure for the lagged equity return variance, exchange rate return variance, and their covariance. The second stage regression is estimated jointly with the zero stage regression by GMM which allows the standard errors of the second stage regression to incorporate not only the uncertainty deriving from the first-stage regression, but also the one from the zero stage regression. The standard errors are based on the Newey-West estimate of the covariance matrix with maximum lag order set equal to T 1/2 . The J-statistic (Hansen, 1982) plus the according p-value are reported for the null hypothesis that the model is well-specified and the moment conditions do hold. The model is estimated for each subsample separately. The extended full sample (May 1987 to August 2011) consists of 280 observations and the second subsample (May 1987 to August 2011) consists of 140 observations. The standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively. A.27

C.8.2 First Stage Regression
A.29  May 2010 -end of sample All episodes except the ones marked by * are taken from Bloom (2009). Bloom identifies periods of major stock market volatility shocks by analysing the deviations of a stock market volatility series from its detrended mean. I partially extend the length of these periods as the events leading to this increased volatility in stock markets already started earlier and lasted longer than indicated by Bloom and because there is evidence in my time series of extensive market reaction. I followed Maggiori (2013) in adding the event of the dotcom bust, and finally included the recent Greek government-debt crisis.

D GARCH -Technical Details and Complete Results
Early specifications of multivariate GARCH (MGARCH) models like the so-called VEC model by Bollerslev et al. (1988) are very general and not convenient when to be put into practice, amongst other things due to the large number of parameters that needs to be estimated. Various, more parsimonious MGARCH models have been proposed to make parameter estimation more feasible while still allowing to capture complex processes for the conditional covariance. A popular specification is the Dynamic Conditional Correlation (DCC) model by Engle (2002). It is based upon the decomposition of the conditional covariance matrix into conditional standard deviations and correlations that are then modelled separately. It is this specification that I employ.

D.1.1 Safety Premium in a MGARCH framework
To see how the theoretical safety premium model above can be estimated by a multivariate GARCH model, note that under complete markets, it should not make a difference whether a home investor shorting the home risk free asset goes long on the foreign risk free asset or on the home market portfolio. In both cases, the expected discounted return should be zero: (A.1) Linearizing the system of equations (A.1) suggests the following expressions for expected excess returns: The first equation in (A.2) corresponds to the already familiar expression for the currency safety premium. The second equation says that the risk premium that the market portfolio return has to offer is equal to the price of risk multiplied by its own conditional variance.
This system of equations can be written in matrix form with xr t+1 standing for the vector of excess returns: where μ t+1 = Cx t+1 and the u t+1 s are an exogenous martingale difference sequence with conditional A.31 variance covariance matrix H t+1 : μ c t+1 is equal to the expected excess return of investing abroad and thus the currency safety premium. The coefficient that corresponds to the price of risk b t and thus the coefficient of interest here is γ c 1 . 23 I will first treat it as constant as I did in the GMM analysis, and later allow it to be time-varying. 24 In the Dynamic Conditional Correlation (DCC) model by Engle (2002), u t+1 is assumed to follow a normal distribution. The conditional covariance matrix H t+1 is decomposed into a matrix of conditional variances D t+1 and a matrix of conditional (quasi-)correlations R t+1 : 25 The conditional variances are modelled as univariate GARCH(1,1) processes: The conditional covariances are modelled as a nonlinear function of these conditional variances. The conditional quasicorrelation parameters that weight the nonlinear combination of the conditional variances also follow a GARCH-like process: being a matrix of conditional quasicorrelations. u t+1 is a vector of standardized residuals: D −1/2 t+1 u t+1 . Given the strong indication of asymmetry in the correlation between changes in the Swiss franc exchange rate and stock market changes and the implied asymmetry in the conditional covariance, I have these models estimated by quasi-maximum likelihood (QML). There is still an active literature developing GARCH models further and suggesting specifications that model in one way or another such asymmetries specifically (see for example Bekaert et al. (2015)). Already the quasi-maximum likelihood estimator, however, allows to estimate MGARCH models consistently without having to worry about about modeling the non-Gaussianity in the shocks (see Fiorentini and Sentana (2007)). 23 I allow for the currency price of risk γ c 1 and the market price of risk γ m 1 to be different as these two prices might reflect some different other factors that are not controlled for. 24 Even tough equation (A.2) does not contain constants, I include them in order to be consistent with the GMM estimation.
25 As stated by Engle (2009) and Aielli (2011), the parameters in Rt+1 are not standardized to be correlations and are thus known as quasicorrelations.

D.1.2 MGARCH results
During the estimation process for the first subsample, I encountered some convergence problems, which are a common issue of GARCH models when put into practice (see for example Silvennoinen and Teräsvirta (2009)), so I only present results for the full sample and the second subsample.
Let us first have a look at the conditional covariances implied by these GARCH models which are pictured in Figure A.12. In the case of the USD, its evolution is pretty comparable to the one estimated with instruments. In the case of the CHF, across all exchange rates and samples it now looks much closer to what I would expect, with clear peaks in crisis episodes. Altogether, the estimates for the conditional covariance implied by the GARCH models seem to be more convincing than the ones calculated with instruments.
The second object of interest are the price of risk estimates (see the first row of Table A .19).
They are all positive, even though insignificant. Overall, they are roughly comparable in magnitude to my three-step GMM estimates when using the zero-stage prediction to measure the expected exchange rate change (recall the values of the last and second to last column in Table 7) and thus support this solution to the measurement error problem. Based on the second period GARCH estimates, the safety premium for the Swiss franc reflected in the EUR/CHF exchange rate would be 2.5% (on an annual basis) on average and reach its maximum of around 12.5% during the recent financial crisis, thus values that are larger than the ones suggested by my GMM results from section 7.3.
Altogether, however, also this GARCH model finds only weak evidence that investors are rewarded for their exposure to currency risk, which is consistent with earlier GARCH literature. De Santis and Gérard (1998) estimate a BEKK GARCH model to find the magnitude of the premium for currency risk based on the international CAPM and only obtain insignificant results when estimating constant prices of risk. 26

D.2 GARCH -time-varying price of risk
The time-varying price of risk in the form of γ c 1,t and γ m 1,t is modelled using a linear function: where the κs are 1x5 vectors. Y t corresponds to a set of instruments including a constant, the market index dividend price ratio dp t and the change in the gap between long-term and short-term interest rates (yield spread) ys t measured by the yield of 10-year government bond in excess of the 1-month interbank rate. Furthermore, it includes the change in the home risk-free interest 26 While the GARCH specifications BEKK by Engle and Kroner (1995) and DCC are shown to produce very similar results (see Caporin andMcAleer (2008, 2012)), the DCC model is computationally more attractive. Figure A. A.34 rate r f,t and the yield difference of Moody's BAA-rated corporate bonds over Moody's AAA-rated corporate bonds baa aaa t (taken from FRED), which is used as a measure for default risk. This way of parametrizing the risk price allows to easily check for the time-variation of the coefficient by setting all κ's except the first one (the one related to the constant) equal to zero.
Due to some convergence problems, I was forced to run the model for the USD exchange rate index with only two out of the four instruments. The results in the first six rows of Table A.20 are somewhat more encouraging than the ones from the three-step GMM procedure. For both the USD exchange rate index as well as for the EUR/CHF exchange rate in the second subsample, there are statistically significant coefficients on some of the interaction terms indicating that the price of risk indeed has a time-varying component. Figure A.13 plots the predicted prices of currency risk and the corresponding safety premiums for these two exchange rates. The risk price coefficients see long periods of negative values, especially in the years between 1998 and 2002, a period with multiple major crisis and therefore hardly a time when investors on average were willing to pay extra for risk instead of asking for a compensation. So altogether, this price of risk picture should probably be looked at with caution. The same holds for the safety premium figure for the USD, while the one for the CHF provides convincing values. Based on these estimates, the safety premium for the Swiss franc reflected in the EUR/CHF exchange rate would be 3% on average (on an annual basis) and around 6% up to 46% during the recent financial crisis. These values are considerably larger than what the GMM results from section 7.3 suggest.
But is it really the case that the time-varying price of risk models should be preferred to the constant price of risk models? The better performance of the models with a time-varying risk price goes with a higher complexity of these models, measured by the number of parameters. Trading off fit against complexity in the GARCH models, both the Akaike and Bayesian information criterion suggest going with the constant price of risk version. Thus, restricting the price of risk to be constant seems still to be justified.  Notes: This table reports the quasi-maximum likelihood estimates of the DCC GARCH model for the case of no time variation in the price of risk. The mean equation relates the excess return xr t+1 of an asset to its risk. xr c t+1 corresponds to the excess return of investing abroad: xr c t+1 = γ c 0 + γ c 1 σ c,m,t+1 + u c t+1 , where γ c 1 corresponds to the price of currency risk. xr m t+1 is the excess return of investing in the home equity portfolio: γ m 0 + γ m 1 σ 2 m,t+1 + u m t+1 , where γ m 1 corresponds to the price of market risk. The vector of shocks u t+1 is ∼ N (0, H t+1 ), where H t+1 is modelled in the style of DCC and decomposed into a matrix of conditional variances and a matrix or conditional (quasi-)correlations. For details see section D.1.1. The estimates are based on monthly returns from January 1990 to August 2011. The number of observations is 259 for the full sample and 152 for the second subsample. The robust Wald test is reported for the null hypothesis that the price of risk coefficients γ c 1 and γ m 1 are jointly equal to zero. Robust standard errors are reported in square brackets. ***, **, and * denote significance levels of 1, 5, and 10%, respectively.