## Introduction

This week, I have spent an undue amount of time on what was supposed to be a simple short study with one of my colleagues, Paritosh Patel, in our London office. I should have learnt, by now, that these things rarely turnout as simple as expected (despite using our eminent tool). What we wanted to do was to give a client some starting ground for a more ambitious FX-project and the following is part of that effort.

In theoretical constructs, exchange rates are normally quite straight-forward with interest rate differentials deciding the price of a currency with respect to some type of no-arbitrage condition. In practice, however, this is not as simple why most economic models posit a number of fundamental variables such as relative money supplies, outputs, inflation and interest rates to explain the swings and roundabouts of currency markets.

As anyone even remotely linked to currency markets can attest to, most (all?) such approaches fail as the FX-gyrations are either bigger or longer than what any mandate would allow (and stomach would endure). Over the past decades, nonetheless, two approaches that have been making progress among FX analysts are the Fundamental Equilibrium Exchange Rate (FEER) and the Behavioral Equilibrium Exchange Rate (BEER). While there are no commonly agreed definitions of these approaches, we briefly define FEER as the exchange rate which in the medium term allows an economy to reach internal and external equilibrium simultaneously. BEER, on the other hand, explains the behavior of the exchange rate with variables that influence the real exchange rate in the long term. Here we will focus on the BEER, which is common for most FX-analysts (pun intended), and developed by Clark and MacDonald (1998). The calculations are using the pound (GBP) as exchange rate of study and a ‘UK-centric G4’ as counterparts.

## Data and calculations

Calculating the **real effective exchange rate** is perhaps the most critical step. Ironically, this is also where researchers often spend the least time. – So have we. And the weights are but one issue; here I have used the BoE Narrow exchange rate index weights to have a base. Ideally you would want to have measures that encompass all types of real and financing flows and not just trade in goods (as is the most common). The variable chosen to deflate the nominal bilateral exchange rates is CPI, but core-CPI, PPI, GDP-deflator, ULC, wages and a number of other measures would probably be as good if not better. In calculations, this variable is expressed in logs. *Nota Bene*, which many seem to stumble on, that the calculated exchange rate has no obvious unit – it is the sum of the weighted logs of a CPI-deflated currency.

**Terms of trade** is the ratio of export and import prices. Here, it is put in relation to the weighted foreign terms of trade. The **relative price** between non-trade to traded is calculated as CPI to PPI. This is also put in relation to a foreign, weighted, measure. Both ‘terms of trade’ and ‘relative price’ are expressed in logs. **Net foreign assets** is total foreign assets minus total liabilities to foreigners in percent of GDP.

The **relative stock of government debt** is government debt to GDP relative a weighted foreign counterpart. **Real interest rates**, finally, is the domestic real interest rate (10 yr Gilts minus CPI inflation) minus a similar weighted measure for China, Euro Area, Japan and the United States.

The unit root test is performed within the VECM-functionality (Johansen) and indicates one cointegrating relationship at the 1% confidence level. As suggested by Clark & MacDonald we also add a constant to the cointegrating relationship.

As you can see when you play around with the model and its settings the equilibrium relationship between the exchange rate and its fundamentals is extremely unstable rendering the equilibrium exchange rate, of course, equally unstable. This is, I think, a quite common problem for BEER:s and, in retrospect, this can perhaps be attributed to the inclusion of China, which is a country with rapidly changing economic structure. That said it probably more rewarding to try to specify the model better, something I gladly hand over to anyone bold enough to look into the machinations of the Macrobond-file.

If nothing else, the resulting equation (for the exchange rate, lq) in our VECM-model can perhaps help our users find their way around the, admittedly, quite opaque statistical output produced by Macrobond (we are working on improving this):

*lq _{t }*= –0.05342Δ

*rr*Δ

_{t-1 }–0.04731*rr*

_{t}_{–2}

*+*0.38568Δ

*q*

_{t}_{–1 }–0.15908Δ

*lq*

_{t}_{–2}–0.77771Δ

*ltot*

_{t}_{-1}–0.06988Δ

*ltot*–0.87222Δ

_{t–2 }*ltnt*+0.51517Δ

_{t-1 }*ltnt*

_{t}_{–2}+0.07890Δ

*nfa*

_{t}_{–1}–2.25265Δ

*nfa*

_{t}_{–2}+0.25434Δ

*gnfl*

_{t}_{–1}+0.57035Δ

*gnfl*

_{t}_{–2}+0.07152(

*ECT*

_{t}_{–1})

Where the loading (alpha) on the error correction term has a suspicious positive sign, and in turn looks like:

*ECT _{t}*

_{-1}= 1.00000

*rr*

_{t}_{-1}– 2.05177

*lq*

_{t}_{–1}+32.3325

*ltot*

_{t}_{-1}

*–14.09955*

*ltnt*

_{t}_{–1}+ 33.97836

*nfa*

_{t}_{–1}–0.81630

*gnfl*

_{t}_{–1}+ 1.26102

Despite most of the parameters (beta) making sense, they are rarely significant, but this is mainly due to the lack of a sufficient number of degrees of freedom as it changes (together with a whole lot of other stuff) – improves – when estimated on a quarterly basis. Now, with all the cautionary warnings and call for further investigations in mind, this is what it looks like: