Using SUR Regression to Study Currency Exchange Rates

A practical RainbowStats example using domestic money velocity and U.S. money velocity

Seemingly Unrelated Regression, or SUR, is useful when several regression equations describe related markets. Each equation may have its own dependent variable and explanatory variables, but the unexplained shocks may be correlated across equations.

Currency markets are a natural example. The Canadian dollar, Brazilian real, Mexican peso, Japanese yen, and Australian dollar may each respond differently to local economic conditions, yet they are also exposed to common global and U.S. monetary shocks.

This example shows how RainbowStats can estimate the individual currency regressions, combine them into a simultaneous system, and then apply the SUR_REGRESSION command.

The Economic Model

For each country, the regression relates the change in the exchange rate to the change in domestic money velocity and the change in U.S. money velocity. Recall in my previous blog I demmonstated how money velocity affects inflation.

Money velocity is calculated as:

The broad-money aggregate differs by country. The goal is not to force every country into a U.S.-style M2 definition, but to use a consistent broad monetary aggregate available for that country.

The RainbowStats Script

The full example can be run in RainbowStats here: Run the RainbowStats script.

GDP_US=GDP
GDP_BR=NGDPSAXDCBRQ
GDP_CA=NGDPSAXDCCAQ
GDP_MX=NGDPSAXDCMXQ
GDP_JP=NGDPSAXDCJPQ
GDP_AU=NGDPSAXDCAUQ

M2_US=MABMM301USM189S
M2_BR=MABMM301BRM189S
M2_CA=MABMM301CAM189S
M2_MX=MABMM301MXM189S
M2_JP=MABMM301JPM189S
M2_AU=MABMM301AUM189S

VEL_BR=Set_Name(GDP_BR/M2_BR,Velocity_for_Brazil)
VEL_CA=Set_Name(GDP_CA/M2_CA,Velocity_for_Canada)
VEL_MX=Set_Name(GDP_MX/M2_MX,Velocity_for_Mexico)
VEL_JP=Set_Name(GDP_JP/M2_JP,Velocity_for_Japan)
VEL_AU=Set_Name(GDP_AU/M2_AU,Velocity_for_Australia)
VEL_US=Set_Name(GDP_US/M2_US,Velocity_for_United_States)

line_Chart(
    same_date_range(
        VEL_BR,
        VEL_CA,
        VEL_JP,
        VEL_MX,
        VEL_AU,
        VEL_US
    )
)

CUR_BR=CCUSMA02BRQ618N
CUR_CA=CCUSMA02CAM618N
CUR_MX=CCUSMA02MXM618N
CUR_JP=CCUSMA02JPM618N
CUR_AU=CCUSMA02AUM618N

List_ALL=Same_Date_Range(
    List(
        CUR_CA,VEL_CA,
        CUR_BR,VEL_BR,
        CUR_MX,VEL_MX,
        CUR_JP,VEL_JP,
        CUR_AU,VEL_AU,
        VEL_US
    )
)

LIST_CA=list(
    ex(List_ALL,Series:0),
    ex(List_ALL,Series:1),
    ex(List_ALL,Series:10)
)

LIST_BR=list(
    ex(List_ALL,Series:2),
    ex(List_ALL,Series:3),
    ex(List_ALL,Series:10)
)

LIST_MX=list(
    ex(List_ALL,Series:4),
    ex(List_ALL,Series:5),
    ex(List_ALL,Series:10)
)

LIST_JP=list(
    ex(List_ALL,Series:6),
    ex(List_ALL,Series:7),
    ex(List_ALL,Series:10)
)

LIST_AU=list(
    ex(List_ALL,Series:8),
    ex(List_ALL,Series:9),
    ex(List_ALL,Series:10)
)

REG_CA=Regression(logdiff(LIST_CA))
REG_BR=Regression(logdiff(LIST_BR))
REG_MX=Regression(logdiff(LIST_MX))
REG_JP=Regression(logdiff(LIST_JP))
REG_AU=Regression(logdiff(LIST_AU))

Simu=SIMU_regression(
    REG_CA,
    REG_BR,
    REG_MX,
    REG_JP,
    REG_AU
)

sur=SUR_regression(Simu)
slideshow(sur)

Why the Script Calculates Velocity Before Applying LOGDIFF

The order of operations matters. The script first calculates the level of money velocity:

VEL_CA = GDP_CA / M2_CA

It then applies LOGDIFF to the complete regression list:

REG_CA = Regression(logdiff(LIST_CA))

This produces the change in the logarithm of velocity:

That is the correct transformation. It measures the growth of nominal GDP relative to the growth of the money stock.

An equivalent alternative would be to calculate GDP growth and money growth separately:

GDP_GROWTH = logdiff(GDP_CA)
MONEY_GROWTH = logdiff(M2_CA)
VELOCITY_CHANGE = GDP_GROWTH - MONEY_GROWTH

This is algebraically equivalent to applying LOGDIFF to the velocity ratio, apart from minor rounding differences. The existing RainbowStats script is simpler and clearer.

Building the Individual Regressions

Each country is estimated separately first. In every list, the exchange rate is the dependent variable, domestic velocity is the first explanatory variable, and U.S. velocity is the second explanatory variable.

REG_CA=Regression(logdiff(LIST_CA))
REG_BR=Regression(logdiff(LIST_BR))
REG_MX=Regression(logdiff(LIST_MX))
REG_JP=Regression(logdiff(LIST_JP))
REG_AU=Regression(logdiff(LIST_AU))

Applying LOGDIFF transforms the exchange rates into approximate percentage changes and transforms velocity into changes in monetary turnover. This is generally more appropriate than regressing exchange-rate changes on trending levels.

Combining the Equations

The five regressions are combined into a simultaneous regression system:

Simu=SIMU_regression(
    REG_CA,
    REG_BR,
    REG_MX,
    REG_JP,
    REG_AU
)

The simultaneous system preserves the individual equations while allowing RainbowStats to compare their residuals and estimate the cross-equation covariance structure.

Applying SUR

The complete system is passed to the SUR estimator:

sur=SUR_regression(Simu)
slideshow(sur)

The output includes the OLS and SUR coefficients, standard errors, t-statistics, residual covariance and correlation matrices, system statistics, and efficiency comparisons.

Main Results

Australia is the only country where domestic velocity also remains significant under SUR, with a t-statistic of approximately −2.1. Domestic velocity is weak or insignificant in the other equations, and Canada's domestic velocity loses significance after applying SUR.

How to Interpret the SUR Gain

The residuals show moderate cross-equation correlation, which supports estimating the currencies as a related system. However, the practical gain from SUR is small.

The system sum of squared residuals is slightly worse, and the average efficiency improvement over OLS is limited. This does not mean SUR failed. SUR is designed to use cross-equation residual covariance to improve estimation efficiency. In this example, the correlations exist, but they are not large enough to materially change most of the coefficient estimates.

Statistical Cautions

• The equation R² values range from approximately 3% to 23%, so most exchange-rate variation remains unexplained.
• Durbin-Watson statistics below 2 in some equations suggest mild positive autocorrelation, which may make the reported t-statistics somewhat optimistic.
• Nominal GDP is quarterly while several broad-money and currency series are monthly. For a strict velocity calculation, monthly money stocks should ideally be converted to quarterly averages before division by quarterly GDP.
• Australia's historical broad-money series contains a major structural break. A later starting date, a break dummy, or removal of the affected observation may be appropriate.
• The regression identifies statistical association, not causation. U.S. velocity should be described as an explanatory factor rather than a proven driver.

Why This Is a Useful SUR Example

This script demonstrates the complete RainbowStats SUR workflow using real economic data:

1. Download nominal GDP, broad money, and exchange-rate series.
2. Calculate domestic and U.S. money velocity.
3. Align all series to a common date range.
4. Estimate each currency equation independently.
5. Combine the regressions with SIMU_REGRESSION.
6. Apply SUR_REGRESSION.
7. Display the complete system with SLIDESHOW.

Conclusion

Across five currency equations, U.S. money velocity is the only variable that consistently explains exchange-rate changes. Domestic velocity contributes independently only in Australia, while Japan shows no statistically significant relationship.

The modest SUR efficiency gains are also informative. They indicate that the currencies share some common shocks, but accounting for those correlations does not dramatically alter the individual regression results.

Most importantly, this example shows how a group of ordinary RainbowStats regressions can be converted into a simultaneous system with only two additional commands:

Simu=SIMU_regression(REG_CA,REG_BR,REG_MX,REG_JP,REG_AU)
sur=SUR_regression(Simu)

That makes SUR_REGRESSION practical for studying related markets, industries, countries, or economic sectors whenever the individual equations may be connected through common residual shocks.

This example is intended as a demonstration of the RainbowStats workflow and not as investment advice.