Disinflation Under Inflation Targeting: a Small Macro Model for Argentina1
1. The Argentine administration that took power in December 2015 has announced a gradual disinflation to single digits by the end of its mandate in 2019. The starting point has been one of the highest inflation levels in the world. Despite the issues in measurement, using a variety of indicators (both public and private) shows that inflation in the city of BA rose sharply from single digits in 2007 and stabilized between 25 and 35 percent (y/y) during 2009 and 2015, with inflation expectations well anchored within that range.2 To a large extent this reflects the earlier administration’s decision to fund the large increase in public sector spending with an inflation tax. Inflation in early 2016 spiked to above 40 percent (y/y using the city of BA CPI index) reflecting the normalization of utilities tariffs needed to phase off energy subsidies, and the sharp depreciation following the removal of ER controls and unification of ER markets in late 2015. Since April 2016, real interest rates have remained positive and core inflation has been on a declining trend, while the economy has moved into recession.
2. To build credibility and align inflation expectations with the new objectives, the government formally launched the inflation targeting framework in September 2016. The intention and timetable to move in this direction was announced early in 2016 and, in May 2016, the BCRA switched to interest rate from money targets as operational variables of monetary policy (Figure 1). In addition, the national statistics agency has resumed publication of credible data on inflation, against which the 12–17 percent target for 2017 will be measured. A companion SIP further describes the operational aspects of the current conduct of monetary policy and the steps that are needed to make further progress on the way to inflation targeting (Jacome, forthcoming).
3. This paper uses a neo-Keynesian type model to analyze the policy challenges and tradeoffs involved in the disinflation process. In embarking on a disinflation process under a new inflation targeting framework, Argentina would need to develop tools for assessing the stance of monetary policy and set policy interest rates. In this paper, we estimate a standard small macroeconomic model, similar to the one used by a number of inflation-targeting central banks for forecasting and policy analysis purposes, and close to the BCRA’s small macroeconomic model (Escudé and others, 2007). Despite its simplicity, and while it abstracts from many important features of the economy, the model captures the key channels of monetary policy transmission and allows discussing the main tradeoffs involved in monetary policy making. In particular, it allows to answer questions such as: How should the path of interest rates look like in order to reduce inflation to a single digit figure in 2019? What is the output cost that could be associated with the disinflation process announced by the authorities? What can moderate this cost?
4. Although the results should be interpreted with caution because of the potential empirical limitations, they suggest that Argentina will likely require significantly high real interest rates and a greater output gap during the disinflation period. Getting to the intermediate target of about 10 percent by 2018 would require about two quarters of real interest rates of about 20 percent, which will then subside rapidly in the following quarters. The cost of disinflation in terms of foregone output, or sacrifice ratio, arises from need to break the prevailing inflation inertia. In its steady-state specification, the model suggests that lowering inflation to the desired 5 percent could cost about 8 percent of potential GDP, a sacrifice ratio of 0.3 (the amount of output lost for every percentage point reduction of inflation). The output cost, however, could be much lower if price setting became more forward looking. In particular, the sacrifice ratio would fall to about 0.2 of potential GDP if the authorities were able to reduce inflation inertia from 70 percent to about 50 percent. We also show that the sacrifice ratio would be lower in our model if the weight on nominal interest rate volatility falls. At the same time, if the authorities were to run monetary policy by trying to reduce the volatility of real exchange rate fluctuations, the sacrifice ratio would fall. Importantly, the model presented in this paper only focuses on the short-term cost of disinflation and is not able to capture its long-run benefits. A large literature has proved that reducing inflation to single digits delivers long-term gains. In particular, lower inflation has been associated to greater market efficiency (Tommasi, 1994), higher private saving (Grigoli and others, 2014), lower income inequality (Albanesi, 2007) and poverty (Cardoso, 1992), and ultimately higher long-term growth (Loayza and Ranciere, 2006).
A. The Model
5. The model is a variant of the new-Keynesian monetary models used in many inflation targeting countries and heavily used within the IMF.3 The key behavioral equations of the model are the following: (1) a Phillips curve equation that relates inflation to past and expected future inflation, the output gap, and the change in the real exchange rate; (2) an output gap (IS) equation that relates economic activity to its own lag values, the real interest rate, the real exchange rate, and the foreign output;4 (3) a risk-adjusted uncovered interest rate parity condition linking domestic and foreign interest rates to the exchange rate, allowing for a stochastic spread differential; a (4) money market equilibrium (LM) equation, which related the demand for real money balances to changes in output and real interest rates, and (5) an interest rate monetary policy reaction. A description of the model is contained in Annex 1.
6. For simulations purposes, we assume that interest rates in the model are set under an optimal monetary policy rule. Although we estimate a policy reaction function to allow identification of other model parameters, the monetary policy reaction function estimated with past data is unlikely to characterize the reaction function of monetary policy under the new IT regime. In the period used for estimating the model a number of shifts in monetary policy frameworks and exchange rate regimes took place. In none of those, nominal interest rates have played an active role, with monetary policy run essentially by closely managing the exchange rate. To better guide policy, in this paper we assume that monetary policy is conducted with the objective to minimize a quadratic loss function that incorporates the main objective of the central bank—to minimize deviation from inflation targets and both output gap and interest rate volatility (Benes 2014). The loss function is of the form
where πt is inflation and Rt is the nominal interest rate. The optimal policy reaction function comes as the solution to the model equations plus a set of equations with the corresponding optimization conditions for minimizing the loss function. For the main simulations of the model, we assume the weights on the output gap, and on the change in nominal interest rates are half those of inflation deviations from target, which deliver sensible model dynamics.
B. Data and Estimation
7. The model was estimated with quarterly data over the period 2003Q1–2015Q4. The period was chosen to allow some distance from the 2001 crisis associated with the end of the convertibility period. Choosing the data for estimation is not trivial in Argentina.
- For the inflation rate, we construct a time series by splicing three different inflation series: the IPC-GBA until 2006, private estimates for inflation in the city of BA from 2006 and 2012, and the inflation index published by the city of Buenos Aires from then onward.
- For GDP, the model uses the revised official GDP time series for 2004Q1–2015Q4 disseminated on June 29, 2016, and spliced with the earlier official statistics for 2003.
- On exchange rates, the estimation uses the official exchange rate. This choice is not straightforward. Parallel foreign exchange markets emerged in 2011 when foreign exchange and capital controls were tightened. We use the official exchange rate, after having tested for both in time series econometric analysis. Although an increasing amount of goods were likely priced taking into consideration the evolution of the parallel market exchange rates, the adjustments in the official exchange rate tended to give rise to large spikes in inflation. Correspondingly, the official exchange rate tends to carry most of the information content. In some alternative specifications, the parallel market spread did carry some limited information content. The parallel market spread, which has de facto been eliminated, is not considered in the final specification.5
- For interest rates, we used LEBAC rates weighted by their maturity structure. Real interest rates are defined as the nominal interest rate deflated by the one-quarter-ahead annualized inflation.6
8. The key parameters of the model have been estimated using Bayesian methods to characterize the structure of the Argentine economy. The estimated parameters and Bayesian priors are presented in Table 1. The values of the key parameters are initially calibrated (i) factoring earlier findings in the literature (ii); to deliver steady state means and standard deviation of the macroeconomic variables that are close to their sample estimates; (iii) to result in plausible estimates of latent variables like the output and exchange rate gaps; and (iv) to deliver well-behaved impulse response functions. All parameters are then re-estimated with the calibrated parameters as priors with varying degree of certainty. The main results obtained are:
- Inflation inertia. The resulting model suggests a significant degree of backward-looking inertia in the inflation process. The point estimate of the parameter of the lagged CPI term in the Phillips curve is 0.67, very similar to what found for Argentina by D’Amato and Garegnani (2009) but higher than estimates in inflation targeting countries in Canales-Kriljenko and others (2009).7 Box 3 in the Staff Report explores the empirical link between inflation inertia and that in wages and inflation expectations.
- Exchange rate pass-through coefficient. The coefficient of the real effective exchange rate on the Phillips curve is 0.1. This is similar to the levels estimated Canales-Kriljenko and others (2009) for Colombia, Mexico, and the European Union, but lower than that estimated for Chile, Brazil, and Peru.
- Effect of output gap on inflation The parameter of the output gap on the Phillips curve equation is nontrivial at 0.472. This is larger than the range of the estimates on the lagged (as opposed to the contemporary) output gap for individual Latin American inflation targeting countries (Canales-Kriljenko and others, 2009). For comparison, D’Amato and Garegnani (2009), using a different methodology, find a very small coefficient of the output gap on the Phillips curve of 0.01.
- Effect of real interest rates on economic activity. The parameter of the real interest rate on economic activity is relatively small (0.1). This low number may reflect the low level of financial intermediation in Argentina, partly due to the prevailing financial repression of the last decade.
As point of comparison, the corresponding parameter for the Latin American inflation targeting countries was 0.16, with parameters among the 9 economies considered fluctuating between 0.1 and 0.2 (Canales-Kriljenko and others, 2009).
|Dynamic IS curve|
|real interest rate gap||b2||0.100||0.09||0.12||Normal||0.10||0.01000||0.095|
|real exchange rate gap||b3||0.040||0.01||0.07||Normal||0.04||0.00500||0.036|
|Foreign output gap||b4||0.600||0.50||0.70||Normal||0.60||0.03500||0.605|
|Interest rate reaction function|
|interest rate inertia (gradualism)||g1||0.700||0.67||0.74||Normal||0.70||0.01050||0.708|
|output gap (gradualism)||g3||4.000||3.90||4.10||Normal||4.00||0.03000||3.998|
|Persistence to shock to inflation target||gtar||0.897||0.85||0.99||Normal||0.90||0.01340||0.891|
|Interest rate increase||delta2||1.600||1.52||1.68||Normal||1.60||0.02400||1.600|
|Expected exchange rate||phi||0.500||0.30||0.70||Normal||0.50||0.06000||0.540|
|Exchange rate inertia||e1||0.500||0.01||0.70||Normal||0.50||0.06000||0.010|
|Real Interest Rate||ss_RR_BAR||0.000||−0.07||0.05||Normal||0.00||0.02500||−0.002|
|Real Interest Rate abroad||ss_RR_RW_BAR||0.020||0.00||0.08||Inverse Gamma||0.03||1/0.04^2||0.008|
|Real interest rate spread||ss_SPREAD||−0.050||−0.10||−0.03||Inverse Gamma||−0.05||0.01000||−0.046|
|Inflation target steady state||ss_D4L_CPI_TAR||0.160||0.10||0.30||Normal||0.20||0.05500||0.216|
- Effect of real exchange rate on economic activity. The estimates suggest that currency undervaluation (a positive real exchange rate gap) stimulate economic activity.8 The real exchange rate appreciation during the convertibility period, in particular, was arguably among the causes for the sharp slowdown of economic activity and increased unemployment toward the end-1990s.
- Output gap inertia. At about 0.7, the inertia in the output gap is fairly high, compared to the average lagged coefficient of 0.48 for Latin America, as estimated in Canales-Kriljenko and others (2009). A high number tends to increase the output cost of monetary policy as economic activity will take more time to get back to its potential level.
- Spillovers. The estimates suggest that developments in trading partners have an important effect on Argentine economic activity.
- Money market. The estimates suggest that a long-run money demand relationship exists and that in the short run real money balances evolve so as to eliminate deviations from such relationship. The long-run trend in velocity was restricted to be close to nil.
- Potential growth. The estimate for the period is of a “steady state” potential growth of 3.1 percent.
- Neutral real interest rate. Given the easy global liquidity conditions prevailing during a large part of the period, especially since 2008, the steady state neutral real interest rates abroad ended up slightly negative during the estimation period. The financial repression supported by exchange and capital controls explains the negative steady state spread of -4.6 percent. Because financial repression and controls have been substantially reduced, this parameter is replaced going forward by the neutral foreign real interest rate in the U.S. (1 percent, Pescatori, 2015) plus an assumed average spread of 200 basis points to account for currency risk associated to higher inflation and volatile financial conditions.
9. We now provide a description of the monetary policy transmission mechanism in the model. An increase in the domestic real interest rates in this model affects inflation through 2 channels. The first one is by slowing economic activity relative to potential. The second is by appreciating the currency. A higher interest rate lowers the interest rate sensitive component of aggregate demand, which tends to reduce the output gap. Also, it increases the exchange-rate and risk adjusted returns of domestic assets encouraging capital inflows and appreciating the currency. In turn, this reduces competitiveness, further slowing economic activity. Thus, although real interest rates do not directly enter the inflation equation in the model, they have a material impact on inflation by affecting the output gap and the real effective value of the currency, which do enter in it. In particular, the inflation equation in the model suggests that inflation in any given period is about two-thirds of the inflation in the earlier period plus about half the size of the output gap and close to 10 percent of the real exchange rate depreciation of the period. This suggests that to contribute to disinflation, the output gap needs to either be negative or become smaller while the currency would need to appreciate, both of which could result from higher real interest rates.
10. Impulse responses provide a feel of the impact of monetary policy on the economy and how the optimal monetary policy would react to shocks to the economy. A shock to interest rates that would deviate from the optimal response by about one percentage point would result in about a 1.2 percentage point increase in real interest rates, a real exchange appreciation of about 0.4 percent in real terms, a slowdown in economic activity that would reduce the output gap by about 0.2 percentage points, and a reduction in inflation of about 0.2 percent, relative to the situation in the absence of a shock (Figure 1). In response to a domestic demand shock that lowers the output gap by about 1 percentage point, monetary policy will react by lowering nominal interest rates by about half a point and real interest rates by about 0.3 basis points. In this situation, the currency will depreciate in nominal and real terms by about 1 percentage point. Inflation would decline by over 0.2 percentage points and slowly recover toward the targeted levels (Figure 2).
Figure 1.Impulse Response to an Interest Rate Shock Figure 2.Impulse Response to a Domestic Demand Shock (to Output Gap Equation)
C. Implications for Monetary Policy
11. Lowering inflation in our model comes with a loss in economic activity, as measured by the sacrifice ratio. In particular, the sacrifice ratio measures the average percent output loss relative to potential for each percentage point reduction in inflation. Following Benes (2014), we estimate the sacrifice ratio by first computing the steady state inflation for a 30 percent inflation target (the level of expected inflation prevailing over the last five years). We then compute another model with an inflation target of 5 percent, as envisaged by the authorities by 2019. Next, we simulate the transition to the new model starting from the steady state of the model with 30 percent and compute the cumulative sum of negative output gaps (Figure 3). This percent output loss is then divided by the 25 percentage point reduction in inflation. The exercise suggests the permanently lowering inflation to 5 percent from 30 percent in 5 years would come at a cumulative output cost of 8 percent of potential output. Thus, the sacrifice ratio would be of 0.3 percentage points of potential output loss for each percentage point permanent decline in inflation.
Figure 3.Argentine Disinflation Path with Optimal Policy Rule
The number is within, but on the low side, of the range of estimates of sacrifice ratio estimated in the literature. Nevertheless, the range of values in the literature is wide reflecting varying empirical methodologies and samples that do not abstract from the possible simultaneous shocks that affect output during disinflation (Anderson and Wascher, Ball, 1994; 1999; Friedman, 1994; Sanchez, Seade, and Werner, 1999; Zhang, 2005). For example, Ball finds that the average sacrifice ratio is 1.4 for quarterly data and 0.8 for annual data for 28 episodes for nine countries. Cross-country comparisons in the literature conclude that the sacrifice ratio is lower when the starting point of inflation is higher. For starting inflation between 20 and 40 percent, Sanchez, Seade, and Werner find an average sacrifice ratio of 0.6.
12. The stylized disinflation simulation also suggests four important observations. First, disinflation has an adverse impact on economic activity, which peaks at about 1 year. Second, it will require high real interest rates, which will coincide with falling nominal interest rates and inflation. Third, it will likely be associated with a real exchange rate appreciation that peaks within the first year before gradually tapering off. Fourth, the disinflation effort would need to be supported by a declining path in money growth rates.
13. Varying a few key parameters of the model may yield a higher sacrifice ratio. In particular, with a lower elasticity of inflation on the output gap (a2) than our estimate, a higher contraction of economic activity would be needed to lower inflation to the desired targets, increasing the ratio. And with a higher elasticity of aggregate demand on interest rates (b2), higher real interest rates would have a more disruptive effect on economic activity, also increasing the ratio. Since staff estimates appears to be on the high side for the first parameter and on the low side for the second, it may well be that the low sacrifice ratio obtained by staff is somewhat underestimated.
14. Lowering the degree of inflation inertia reduces the sacrifice ratio. The adoption of inflation targeting may tend to reduce the amount of inflation inertia. For the aggregate of five Latin American inflation targeting countries, Canales-Kriljenko and others (2009) found that the backward looking component of inflation was smaller than the forward looking component, a feature that was even more pronounced in the advanced economies mentioned in their study. The literature on endogenous credibility suggests that the output cost could fall significantly if the inflationary inertia could be reduced (Isard, Laxton, and Eliasson, 2001; Argov and others, 2007; Ali and others, 2009). To explore the gains coming from the expectations channel we perform a sensitivity analysis to the inflation inertia coefficient (a1) of the Phillips curve. A fully backward looking inflation dynamics (with an a1 parameter of 0.99) would result in almost twice as much the sacrifice ratio than the one described above (Figure 4). On the other hand, with a parameter value of 0.5, the short-term output cost of disinflation would fall to 6 percent of potential output, implying a sacrifice ratio of around 0.2.
Figure 4.Sensitivity of Potential Output Cost of Disinflation (Cumulative Output Gap) to Different Degrees of Inflation Inertia
15. The sacrifice ratio also depends on the monetary policy authorities’ relative tolerance for output gaps and inflation deviations from target. In particular, and as it would be easy to expect, the sacrifice ratio falls as the relative weight of the output gap increases. Increasing the weight on interest rate volatility in the loss function, on the contrary, increases the sacrifice ratio. Such a higher weight may be motivated by concerns about the real sector effects of financial sector volatility, and the build-up of vulnerabilities that could result in costly disruptions in economic activity associated with financial crisis. While lower volatility in interest rates may have financial sector benefits, it may also have a cost in terms of economic activity during disinflation, as illustrated in Figure 5. This is because adjusting nominal interest rates more slowly during disinflation may result in higher real interest rates and a more appreciated currency, with an adverse impact on economic activity. Nevertheless, the literature warns that the effect of interest rate smoothing on the sacrifice ratio may depend on the specific form in which interest rates enter the authorities’ loss function (Woodford, 2002). Once disinflation is achieved, attempting to reduce nominal interest rate volatility would tend to result in greater volatility in output and inflation (Honjo and Hunt, 2006).
Figure 5.Sensitivity of Potential Output Cost of Disinflation (Cumulative Output Gap) to Weights in Loss Function
16. The sacrifice ratio in the model falls if the authorities try to reduce exchange rate volatility. Argentina’s policy makers have been concerned about exchange rate volatility, at least to judge by their history of heavy foreign exchange intervention and exchange rate regimes with little exchange rate flexibility. To analyze the effects of these considerations on the monetary policy tradeoffs, we include in the loss function a concern for the real exchange rate volatility.
where lzt is the real exchange rate (in logs). In the context of perfect certainty such as the one envisaged in this paper, a concern about real exchange rate volatility can be justified based not only on the direct impact that the real exchange rate can have on economic activity and inflation, but also on its indirect impact, through the value of financial wealth in a highly dollarized economy. The corresponding optimal monetary policy rule implies that the monetary authority gets to influence both the exchange rate and interest rates, and that both exchange rate and interest rate policy decision respond to each other. Interest rates may rise to smooth exchange rate fluctuations, especially on occasions in which this helps achieve the other policy objectives. In the simulation, a higher weight on exchange rate volatility results in a smoother path for real exchange and interest rates, and a lower output gap during the transition to lower inflation (Figure 6). This is because in our model the real exchange rate plays a crucial role for both inflation and economic activity. Because the effect of the exchange rate prevails over that of the output gap in the inflation equation, a policy rule with no weight on exchange rate volatility would tend to rely more on real appreciation to reduce inflation at the expense of a stronger impact on economic activity, resulting in a higher sacrifice ratio (Fischer, 1984). Along these lines, an additional instrument to influence the behavior of real exchange rates would arguably help in the authorities’ objective of maintaining price inflation stability while minimizing the volatility in output fluctuations.
Figure 6.Simulations of Disinflation with Real Exchange Rate Volatility in Loss Function Varying Weights on Exchange Rate Volatility (lambda3)
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A. Phillips Curve
The Phillips curve captures the short-term tradeoff between economic activity and inflation. When economic activity exceeds potential GDP, that is when the output gap (lgdp_ gapt) is positive, it exerts inflationary pressure. The opposite is true when economic activity is below potential. But the curve also recognizes that inflation typically has significant inertia, and only partially reflects expectations of future inflation. The Phillips-type curve in this model also captures the fact that real exchange rate movements
B. Dynamic IS Equation
The dynamic IS equation captures the relationship between economic activity and monetary conditions as well as spillovers from trading partners. Higher real interest rates (rr_ gapt) lower economic activity while more depreciated currencies in real terms (lz_ gapt) tend to boost it. Real interest and exchange rates are measured relative to their equilibrium, time-varying levels. In addition, the equation encompasses the view that economic activity in trading partners have a direct impact on domestic economic activity through the foreign output gap (lx_ gdp_ gapt), after recognizing that trading partners may have different long-term growth rates associated to among other things different demographic structures and dynamics. The dynamic IS equation reckons economic activity tends to move gradually and has a nontrivial backward looking component. Finally, the error term includes demand shocks (independent from monetary conditions, inertia, and foreign economic developments) that have an impact on economic activity. Autonomous fiscal developments would appear as shocks to domestic demand.
C. Uncovered Interest Rate Parity and Exchange Rates
The model includes an uncovered interest rate parity equation expressed in real terms. It states that real interest rate differentials tend to be associated to changes in the real exchange rate in the long run, although short-term deviations (shocks) do take place. The real interest rate differentials are relative to each country’s equilibrium real interest rate levels.
The model allows expectations of the real exchange rate to have a backward and a forward looking component. A sovereign spread is added to the version after estimation, to recognize the sharp relaxation in financial repression.
Nominal and real exchange rates are linked by the following identity:
D. Policy Reaction Function
The model includes a policy reaction function to set nominal interest rates. The equation is a crucial element going forward as it would guide the central bank in setting its operational variable. Nevertheless, it could be a controversial element for the estimation part because clearly Argentina did not explicitly target inflation or use interest rates to achieve that objective. The analysis will later include different policy reaction function, including by highlighting the policy frontier that depends on weights to a loss function in terms of inflation and economic activity.
The policy reaction function standard in FPAS-type models implicitly argues that to stabilize inflation to a given target, without undue disruptions in economic activity or financial distress, the central bank should set nominal interest rates (Rt) around a neutral level associated with the prevailing real interest rate(rrt) and forward looking inflation
E. Money Market
Given the importance of the need for seigniorage to explain the relatively high level of inflation prevailing over the last decade, the model includes equations describing the behavior of monetary aggregates. Money has been included in GPM-type models in the cases of Kenya (Andrle and others, 2013) and Uruguay (Portillo and Ustyugova, 2015). For Argentina, Basco, D’Amato, and Garegagni (2011) find that the introduction of the real money gap does not improve forecast accuracy for quarterly data, but it significantly adds predictive power at lower frequencies.
Following Portillo and Ustyugova (2015), the model in this paper includes an error correction specification for the evolution of the money aggregate M2. Changes in real money balances
Shocks to money demand can be persistent and add a trend to the level of real money balances.
Prepared by Jorge Ivan Canales-Kriljenko.
Large discrepancies between official inflation statistics at the national level and private sector and provincial estimates suggests significant uncertainty about the measurement of inflation since late 2006. This study uses a consumer price series concatenated from private and provincial sources as later described.
See, for example, Berg, Karam, and Laxton, 2006a and 2006b; Carabenciov and others, 2008; Canales Kriljenko and others, 2009. A similar model was also estimated in a BCRA paper (see Elosegui, Escudé, Garegnani and Sotes Paladino, 2007.)
These variables are expressed in gap terms, that is, as deviations from their steady state values or long-run trends.
Although excluding the parallel market rate from the specification could in principle bias the results under conventional econometric methods, the use of Bayesian priors and methods would tend to reduce this bias.
In particular, the natural logarithm of (1+nominal interest rate) less four times the one-quarter ahead difference in the natural logarithm of the price level.
In the BCRA model estimated by Elosegui and others (2007) with data covering the period 1993–2006, the weight of past inflation is slightly higher than the weight on future inflation.
This is consistent with what found in many other studies covering Argentina and other EMs in general, including Berg and others (2008), Glüzmanna, Levy-Yeyati and Sturzenegger (2012), Hausmann and others (2005), Levy-Yeyati and Sturzenegger (2007), Rapetti (2011), Rapetti, Scott, and Razmi (2011), and Rodrik (2008).
In all that follows, l before the variables means the variable is in natural logarithm, while the dot means the variable is in log-difference. The dot4 means log difference four quarters apart. Nominal interest rates (R) represent the natural logarithm of gross interest rates, for example, ln(1+R/100).