A Three Variable Macro Model : Estimation

We take simulated data from the three variable macro model in the last blogpost and try to estimate the parameters using Structural Vector Auto-Regression (SVAR). 

1. Graph, Summary Statistics & Correlations

Inflation is the annual increase in general prices measured in percentages. Nominal interest rates are also measured in % per annum. Output gap is usually calculated using some statistical filter (thus it has very small values). Typically a positive output gap implies that the economy demands for goods and services over and above its capacity. The quarterly data looks like this: 

Visually, the graph shows that inflation, interest rate and output gap are stationary. Stationarity means that the series is likely to have constant mean- Inflation always seems to return to 4%, interest rates to 6% and output gap to 0. Further, each series seems to have a constant variance - fluctuations do not get wider or narrower but remain even throughout.

We now look at the correlation between variables and their lagged values: 

Lets look closely at this matrix - each (i,j) entry in this matrix tells us the correlation of the i^th row entry with the j^th column entry. All variables are highly correlated with their own lags, indicative of strong persistence. Inflation & interest rates go hand in hand, while output gap moves against both of these variables. 

We go to our central bank’s website and find that it’s constitution says that “…our main endeavour will be to keep the general rise in consumer prices to 4% per annum…”. This means that the central bank was targeting inflation to be at 4%. The summary statistics confirm that inflation has been largely kept at the target of 4%. Further, nominal interest rates have been about 6% on average, which indicates that average real interest rates have been about 2%. This should correspond to the “natural” level of real interest rates - consistent with zero output gap. 

B. Structural Vector Auto Regression

Let us assume that the “structural” model is given by: 

The deltas form the constants in the regression, the gammas form the contemporaneous effects and the rest of the coefficients tell us how the variables are connected via lags. The “structural” shocks are independent of each other also. We cannot estimate the VAR in this form, we bring it to its “reduced form”:- 

After estimating the “reduced form” we need to go back to the “structural form”. This is the identification problem. If we compare equation (1) and (2) we see that: 

“Reduced form” shocks are not structural shocks but rather mixtures of structural shocks. There are many possibilities - output gap, inflation & interest rates all can effect each other contemporaneously - 6 free parameters (gammas). 

When we move from the estimated VAR parameters to the SVAR we need to restrict a few parameters to achieve identification. In this case we impose restrictions on at least 3 out of 6 of these contemporaneous effects. One plausible restriction is the Keynesian idea that firms don’t like to change prices that often and so inflation cannot respond immediately to output & interest rates. And sluggish output too does not respond to changed interest rates. This also fits well with the Monetarist idea that monetary policy suffers from transmission lags. In contrast, we allow inflation to effect output & interest rates without lag and output to effect interest rates without lag. So we impose: 

After this we deal with the question of how many lags. More lags means better fit but more parameters. The goal is to explain more with less. These criteria give us the lag order for which there is maximum goodness of fit with least parameters. The results say that a lag order of 1 makes most sense. 

We estimate the SVAR:

 Overall the model fit is excellent. But all contemporaneous effects are insignificant at 5 % level. Some lagged variables are insignificant at 5 % level. All those that were insignificant are force to zero in the final estimation. 

  The estimates change slightly, and one term that was earlier significant becomes insignificant (the effect of lagged output gap on interest rate becomes insignificant). The final estimates of parameters are given here:  

Further we can also say that inflation has a unit root. This corresponds well to the idea that expectations are adaptive. The final estimates also show that the central bank is indeed a pure inflation targeter. It does not seem to respond to output fluctuations. 

D. Impulse Response Function (IRFs)

The next step is to generate Impulse Response Functions (IRFs). These IRF plots tell us what structural shocks do to the economic system: 

1. Top row - positive ‘supply’ shock - purely to inflation: Pushes inflation above target immediately - the central bank responds a quarter later by raising interest rates. Rising inflation expectations, lowers real interest rates, which lead to a temporary positive output gap but that is quickly reversed because nominal interest rates have been raised. 
2. Middle row- positive ‘policy’ shock - purely to interest rates: leads to an immediate hike in interest rates which is quickly reversed, but which causes demand to fall in the next period and reduce inflation after another quarter. Interest rates are reversed, and through demand, bring inflation back up to target
3. Third row - positive demand shock - purely to output: which raises demand immediately. This has effect on inflation in the next quarter, which rises. And then interest rates rise in the 3rd quarter, to correct the rise in inflation.

Verdict

So did we get it right? Yes. All the coefficients were accurately identified, in terms of both direction and magnitude. And the Impulse Response Functions approximated the simulations carried out in the previous blog post. Just compare - 

The True Model :   

While this exercise was done in a very controlled environment we have demonstrated that advanced estimation techniques like the SVAR do work.

STATA codes to replicate, keep this irf file in working directory. 

A Three Variable Macro Model : Simulation

This blogpost shall simulate a simple 3 variable macroeconomic model of the business cycle. The first variable is the annual increase in general prices or inflation. The second is the “output-gap” which measures the log-difference between actual aggregate production and potential/full capacity GDP. It can also be thought of as a measure of excess demand for goods and services at the aggregate level. The third variable is the interest rate, which is the price of loans and credit in the economy, for both households and firms. 

These three variables are contenders for the most important variables in macroeconomics and influence each other over the business cycle. By simulation, I mean that I will run a virtual experiment. The system will start at rest, and I will introduce different types of ‘shocks’ at various locations, and then observe the dynamic movements of these three variables due to these shocks.

Since this is a purely imaginary economy, I make very simplifying assumptions and am in complete control over what is going on here. STATA Codes to replicate are at the end.

The Model

The following 5 equations fully describe the economy. One unit time in this tiny model economy is a quarter. 

Here (1) is the Aggregate Supply (AS) curve which says that inflation is a function of inflation expectations, output gap and supply shocks. Firms observing excess aggregate demand for goods and services will raise prices in the next quarter. Workers’ expectations of inflation influences how they negotiate for wages; higher the inflation expectations, the higher the wage negotiated. This leads firms to raise product prices. Supply shocks cover unpredicted changes in inflation due to oil prices movements, monsoon & climatic conditions in agriculture, etc.   

(2) is the dynamic Investment-Savings (IS) schedule that says that ex-ante real interest rates effect output gap with 1 quarter lag. Ex-ante real interest rates is defined in (3) and are nothing but interest rates in the same period minus inflation expectations for the next quarter; and measure the real price of credit & borrowing. 

If State Bank of India offers a home loan today, for an interest of 5%, a 10 lakh home will cost half a lakh in interest above the instalments for the principal. But if prices rise 5% tomorrow (and consequently your wage rises 5% with it), you will effectively end up paying zero interest for that loan! Therefore the ex-ante cost of credit depends on the interest rate today and the expectation of inflation tomorrow.

When the expected cost of credit is low consumers will execute plans take a home or car loans, investors and industrialists will take loans to invest in businesses. Therefore spending on goods and services will exceed the optimal capacity of the economy. However, transmission from interest rate to demand takes a full quarter.  

(2) Also says that at some price of credit (r*) the excess demand is zero. This is the “natural” rate of real interest rate, consistent with the economy’s capacity to produce. Apart from ex-ante real interest rates, we also have 𝜂_t which is the demand shock. This could consist of unpredictable changes in spending due to varying preferences for savings, changes in wealth caused by variations in asset prices, exchange rate fluctuations causing adjustments in exports & imports, abrupt movements in consumer and investor sentiment, etc.  

(4) is a Taylor rule, which describes how the central bank behaves. In our imaginary economy, the central bank’s sole mandate is to try and keep inflation at a target (𝜋*). It has control over the interest rate, because it is the banker to all commercial banks and at what rate it lends to them governs how they price credit. 

When inflation is on target (𝜋_t=𝜋*), the central bank keeps interest rates at r* + 𝜋*, so that (given inflation expectations are also equal to target), the ex-ante real interest rate is at its natural levels - and thereby there is no excess demand in the economy. If inflation is below its target, then the central bank lowers the interest rate below r* + 𝜋*. This raises the real interest rate, and according to (2) will create excess demand for goods and services. Excess demand will then, according to (1), raise inflation. The central bank will keep interest rates low until inflation is back on target. 

Apart from this, interest rates also changes due to policy shocks (v_t) which can arise due to frictions in transmission of monetary policy through commercial banks, or simply if the central bank decides to break from its rule of strict inflation targeting.  

(5) tells us that households, investors and workers always expect inflation to be what it was a quarter earlier. This is a very rudimentary rule of thumb, because obviously inflation will not be what it was a quarter earlier, because of changes in demand (whether caused by the central bank or by demand shocks) and supply shocks. Thus agents overlook the behaviour of firms and the central bank in determining inflation outcomes. But this is a simplifying assumption.

In summary: Monetary policy, is entirely concerned about inflation, effects aggregate demand through interest rates with one period lag and demand pressure generates inflation one period later. All three variables are interconnected, across time, to each other. And shocks intrinsic to each of them (demand, supply, policy) generate or power the dynamics in the economic system. 

Furthermore, we can describe the economic system in a more concise manner by solving (1) - (5) in terms of (𝜋_t,R_t,y_t)' the state vector, and writing in Vector notation: 

Stability of the system occurs when the state variables (𝜋_t,R_t,y_t) always tend to return to their steady-state values (𝜋*, r* + 𝜋*, 0) regardless of the magnitude of the shocks. This is assured only for certain ranges of parameter values (β, ϕ,θ), in all other cases the shocks whack the state variables permanently away from their steady state values. Stationarity occurs when the coefficient matrix of the lagged state vector is such - that it’s eigenvalues are within -1 and 1. We ensure this in the simulation. 

Simulations: Multiple Scenarios

We set the following parameters to be the following for the baseline case. The initial values for state variables are given to be their steady state values.

1. Demand, Supply & Policy Shocks 

First I introduce a positive supply shock of unit magnitude at quarter 5, which falls by 0.25 units every consecutive quarter till it reaches zero. What happens to the economy? The economy begins at steady state. Then in quarter 5, the supply impulse hits inflation, which rises above target. This has two effects in quarter 6. First, inflation expectations for quarter 6 rise. So anticipated real interest rate in quarter 6 falls. Thus demand rises in quarter 6, albeit temporarily. Second, the central bank responds to inflation overshooting target in quarter 5, by raising interest rates in quarter 6. This reverses the temporary increase in demand and causes a contraction in demand. This contraction in demand reduces inflation and brings it back to target levels. A negative supply shock, would reduce inflation first, then interest rates would fall after which demand would rise - restoring inflation back up to target levels. 

  

Next I introduce a positive demand shock at quarter 5, which also dies out like the  supply impulse. In this situation, it is demand which rises first in quarter 5 (a boom). This is followed by rising inflation in quarter 6, which has two effects - one, it further bolsters demand by raising inflation expectations and lowering real interest rates and two, it forces the central bank to swing into action in quarter 7. Interest rates thus respond last in this schemata, but are pivotal in bringing the economy back to target. A negative demand would do the reverse: lower demand, then lower inflation, which would have the central bank lower rates, which would reverse the recession and deflation.  

A similar positive policy shock is generated at quarter 5. So the central bank raises rates from in quarter 5, which it only gradually reduces. This reduces demand in quarter 6, which in turn, reduces inflation at quarter 7. Now the central bank must quickly reduce interest rates to bring inflation back up to target.

B. A Dovish Central Bank

The parameter ϕ tells us how hard the central bank reacts to inflation deviations from target, therefore it is a measure of how dovish or hawkish the central bank is. We reduce the value of ϕ to 1.2 and re-simulate the impulses. The key differences (a) the state variables, once disturbed, take much longer to get back to long run equilibrium. (b) the dovish central bank, by responding weakly, allows inflation to climb well above target.

   

C. Disinflation: Gradualist or Cold-Turkey

Here we simulate two different strategies to bring inflation to a lower target level. The goal of disinflation is to bring inflation down from 4% to 2%. The first approach is mild (call it “Gradualist”) while the other is aggressive (call it “Cold-Turkey”). 

 The Gradualist approach seeks to first reduce the target to 3% from quarter 5 onwards and then further reduce the target to 2% from quarter 25. I also keep the the central bank dovish by setting ϕ = 1.2. Cold-Turkey is when a hawkish central banker goes for immediate dis-inflation. In our simulation I re-set ϕ = 1.5 and have the central bank immediately begins to target inflation at 2% from quarter 5 onwards.   

What are the main differences? Gradualism takes a lot of time. Interest rates are not raised very high. It also does not push aggregate demand far below potential. In contrast, Cold-Turkey goes for heavy rate hikes that cause recession. This recession is used to disinflation the economy. Due to adaptive expectations, Cold-Turkey ends up creating serious output loss but achieves the task quicker than Gradualism. 

D. What a Dynamic Stochastic Economic System can look like

I re-simulate, bombarding the economy with random disturbances. The baseline parameters have been taken. The supply, demand and policy shocks that I have given are independent, identical and normally distributed with zero mean and certain variance. 

The resulting dynamics are complex and fascinating - shocks push the system away from its equilibrium but causal forces keep the system stable. 

Disclaimer - This is a purely imaginary economy. It borrows plot elements from the real world and plays out like a Bollywood movie. Any reference to any person or phenomenon, living or dead, is purely accidental and solely expositional!  

Why Simulate? 

1. it is expositional, gives clarity and visualisation
2. tells you what effect each parameter or shock has on the systems dynamics
3. since we built it, we can do whatever we like - we are the omnipresent and omniscient gods - we can explore ideas! 
4. is fun
5. helps test the power of estimation techniques - if estimation techniques are unable to figure out the parameters that were used to generate simulated data then they are probably not very useful against real world data

The next blogpost will cover (e) where we will using only this very simulated data, and see if empirical techniques called the Structural Vector Auto-Regression (SVAR) can get us back to square one i.e where we started from! 

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