Macro-Financial Uncertainty

# Macro-Financial Uncertainty

We provide indices of Macro-Financial Uncertainty for 33 countries (and combinations thereof) on 5 continents.

## Introduction

Our uncertainty measure is based on a simple test statistic that varies with deviations from historical mean and correlation patterns for all given variables. The methodology is in the spirit of the financial turbulence measure proposed by Kritzman and Li (2010) (based on previous work by Chow et al. (1999)) and was modified by Stöckl, Hanke, and Angerer (2017) and Stöckl and Hanke (2014), who also highlight its advantages over other commonly used measures of financial risk and uncertainty. The modifications include the introduction of a weighting scheme that allows to aggregate various variables across various economies into a global measure of Macro-Financial Uncertainty. At the same time the construction of the measure allows to quantify the contribution of individual input factors to find the drivers of current levels of uncertainty.

## Methodology

The intuition behind the uncertainty measure can be illustrated by a simple $$t$$-statistic: Assuming the variable of interest is the change in the short-term interest rate $$\Delta INT_{t}^{ST}$$ and we are interested in the number of standard deviations today’s change deviates from its historical mean: $$\frac{\Delta INT^{ST}_{t}-\mu_{\Delta INT^{ST}_{}}}{\sigma_{\Delta INT^{ST}_{}}}$$. Assuming a reasonable number of observations (e.g., 60 months) for the calculation of $$\mu_{\Delta INT^{ST}_{}}$$ and $$\sigma_{\Delta INT^{ST}_{}}$$, a magnitude above $$2$$ is a $$5\%$$ event (given the $$t_{59}$$ distribution of the statistic) and would happen roughly once every two years. In a macroeconomic context, however, not only short-term interest rates but also changes in long-term interest rates, $$\Delta INT^{LT}$$, may be relevant. A natural extension of the above is to calculate

$\sqrt{\frac{(\Delta INT^{ST}_{t}-\mu_{\Delta INT^{ST}_{}})^2}{\sigma^2_{\Delta INT^{ST}_{}}}+ \frac{(\Delta INT^{LT}_{t}-\mu_{\Delta INT^{LT}_{}})^2}{\sigma^2_{\Delta INT^{LT}_{}}} }.$

This approach, however, neglects the fact that short- and long-term interest rates are usually correlated. The multivariate generalization of the $$t$$-statistic developed by Hotelling (1931), and therefore often called Hotelling’s $$T$$-squared statistic, takes into account fluctuations in both measures as well as their direction:

$\sqrt{\left(\Delta INT_{t}-\mathbf{\mu}_{\Delta INT_{}}\right)^{'}\mathbf{\Sigma}^{-1}_{\Delta INT_{}}\left(\Delta INT_{t}-\mathbf{\mu}_{\Delta INT_{}}\right)},$

where $$\Delta INT_{t}$$ gives the vector of current changes in long- and short-term interest rates in relation to their vector of means $$\mathbf{\mu}$$, given their covariance matrix $$\mathbf{\Sigma}$$. In this case, assuming a high positive correlation of both variables, the measure would indicate a low-probability event, in the domain of approximately $$5\%$$, given that both interest rates move one respective standard deviation in opposite directions. Hotelling’s $$T$$-squared statistic increases in two dimensions: (1) if one realization deviates from the mean and (2) if the comovement in the contemporary variables deviate from their long-term correlation pattern.

## Macro-Financial Uncertainty

In Gächter, Geiger, and Stöckl (2020), we calculate global Macro-Financial Uncertainty for the G7 economies (the United States, Japan, Canada, and the United Kingdom, as well as the Eurozone - summarizing Germany, France and Italy) from the following macroeconomic and financial variables:1

• CPI: year-on-year growth rates of consumer price indices
• INT2Y and INT10Y: yields on 2- and 10-year government bonds (monetary policy)
• REER: real effective exchange rate (international competitiveness)
• STOCKS: fluctutaions in risk premia
• VIX: fluctuations in financial market risk

All variables are entered as (log) first differences. To calculate a global measure of Macro-Financial Uncertainty, we combine all inputs weighting them by the respective GDP of each country (region), $$w_{c,t}$$. To ensure that the index maintains the distributional properties of the $$T$$-squared statistic, we follow Stöckl, Hanke, and Angerer (2017) and normalize the weights by their squared sum. Therefore, we define a diagonal matrix of adjusted weights as $$\mathbf{w_t^{*}}=diag\left(\frac{\mathbf{w}_t}{\sum_{j\in \mathcal{S}} w_{j,t}^2}\right)$$, where $$\mathbf{w_t}$$ is the vector of GDPs and $$\mathcal{S}$$ is the set of all input variables with cardinality $$N=30$$. Our index of global Macro-Financial Uncertainty is thus defined as:

$MFU_{t,\text{global}}:=\sqrt{ \Big(\mathbf{w_t^*}\big(\mathbf{\bar{x}_{t}}-\mathbf{\mu}_{x}\big)\Big)^{'} \mathbf{\Sigma}^{-1}_x \Big(\mathbf{w_t^*}\big(\mathbf{\bar{x}_{t}}-\mathbf{\mu}_{x}\big)\Big) },$

where $$\mathbf{\bar{x}_{t}}$$ is a short-term moving average of inputs at time $$t$$. For all variables that are not available from the beginning of the sample, we adjust the measure accordingly. For the in-sample version of the measure (black line) we use full sample statistics for the calculation of $$\mathbf{\mu}$$ and $$\mathbf{\Sigma}$$, whereas for the out-of-sample version we use a recursively growing window with a starting window of 120 months and let variables that have shorter time series only enter into the calculation once more than 60 observations are available.

## The App

In this App, one can choose to calculate uncertainty measures for individual economies or (GDP-weighted) combinations thereof. The initial selection of variables comprises the full-set of variables chosen by Gächter, Geiger, and Stöckl (2020) but can be subsequently reduced on the left-hand side. In-sample calculations (black line) are based on full-sample moments calculated for the time-frame chosen in the left panel, whereas the out-of-sample index (red line) is initially based on a recursively growing window starting with 120 observations. In the advanced settings, the user can choose a lookback window (short-term moving average to smoothen the currently observed values across the last months for in- and out-of-sample versions of the measure), as well as a rolling rather than a recursively growing window, additionally setting the initial estimation window for the out-of-sample measure.

The next two tabs show the contributions of individual variables (as well as combinations thereof) to the measure of Macro-Financial Uncertainty, calculated using the following relation: $$a'\Omega^{-1}a=\sum_{j=1}^{K} a'\Omega \circ a'=\sum_{j=1}^{K} \left(a'\Omega_{-,1}\: a_1,\dots,a'\Omega_{-,K}\: a_K\right)$$, where $$a$$ is some $$K\times 1$$-vector, $$\Omega$$ is some $$K\times K$$ matrix and the elementwise multiplication is denoted by~$$\: \circ$$. In turn, each $$a'\Omega_{-,j}\cdot a_j$$ is the contribution of element $$j$$ to the overall measure. In the next tab, we follow the alternative approach suggested by Garthwaite and Koch (2016) in determining the individual contributions of each variable. Last, we provide the sourcing information for the data used, most importantly the symbols we used to retrieve data from Refinitiv Datastream, as well as the possbility to download all the data generated by our algorithms. All calculations are carried out using my R-package uncertaintymeasure available on github.

## References

Chow, George, Eric Jacquier, Mark Kritzman, and Kenneth Lowry. 1999. “Optimal Portfolios in Good Times and Bad.” Financial Analysts Journal 55 (3): 65–73. https://doi.org/10.2469/faj.v55.n3.2273.

Garthwaite, Paul H., and Inge Koch. 2016. “Evaluating the Contributions of Individual Variables to a Quadratic Form.” Australian & New Zealand Journal of Statistics 58 (1): 99–119.

Gächter, Martin, Martin Geiger, and Sebastian Stöckl. 2020. “Credit Intermediation and the Transmission of Macro-Financial Uncertainty: International Evidence.” Journal of International Money and Finance forthcoming (February). https://doi.org/10.1016/j.jimonfin.2020.102152.

Hotelling, Harold. 1931. “The Generalization of Student’s Ratio.” The Annals of Mathematical Statistics 2 (3): 360–78. https://doi.org/10.1214/aoms/1177732979.

Kritzman, Mark, and Yuanzhen Li. 2010. “Skulls, Financial Turbulence, and Risk Management.” Financial Analysts Journal 66 (5): 30–41.

Stöckl, Sebastian, and Michael Hanke. 2014. “Financial Applications of the Mahalanobis Distance.” Applied Economics and Finance 1 (2): 71–77. https://doi.org/10.11114/aef.v1i2.511.

Stöckl, Sebastian, Michael Hanke, and Martin Angerer. 2017. “PRIX - A Risk Index for Global Private Investors.” The Journal of Risk Finance 18 (2): 214–31. https://doi.org/10.1108/JRF-09-2016-0118.

1. These six variables were chosen to cover the most important areas of (short-term) risk and uncertainty. The inclusion of additional variables (e.g., short-term proxies for economic activity) did hardly change the corresponding macroeconomic uncertainty indices.