Graduate Macro Theory II: Notes on Quantitative Analysis in DSGE Models

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1 Graduate Macro Theory II: Notes on Quantitative Analysis in DSGE Models Eric Sims University of Notre Dame Spring 2011 This note describes very briefly how to conduct quantitative analysis on a linearized DSGE model. Here we take the very simply stochastic growth model as the benchmark model. 1 The Model and its Solution The first order conditions of the stochastic growth model, linearized about the non-stochastic steady state, are as follows: ( ( βα(α 1k α 1 βαk α 1 c t = c t+1 kt+1 ã t+1 (1 σ σ ( ( c kt+1 = α 1 1 ã t c t + kt (2 β ã t = ρã t 1 + ε t (3 The first equation is the dynamic consumption Euler equation; the second is the capital accumulation equation; and the third is the exogenous stochastic process for TFP. Variables with super-script denote steady state values. Using the notation introduced previously, there are n = 1 control/jump variables and m = 2 state/exogenous variables. In vector notation, we have, where x 1 is n 1 and is the vector of jump variables (in this case a scalar and x 2 is a m 1 vector of state variables: x 1,t+1 n 1 E t x 2,t+1 m 1 (n+m 1 = M x 1,t n 1 (n+m (n+m x 2,t m 1 (n+m 1 We can solve for the policy function as described in previous notes. Let s denote this as: 1

2 x 1,t = Φ x 2,t (4 n m 2 Quantitative Analysis Quantitative analysis involves the (i choosing parameters; (ii constructing impulse responses to structural shocks; (iii doing a variance decomposition; (iv simulating out data from the model and comparing the properties of the simulated data (typically second moments to second moments from the actual data. We will talk more seriously about parameterizing the model later when the model gets more realistic. Loosely speaking, the tradition is to calibrate the parameters of the model. Calibration implies picking parameter values that imply steady state features of the model which align well with actual data. The aggregate production function is: y t = a t k α t n 1 α t In a decentralized version of the model, with competitive markets labor is paid its marginal product. Hence total payments to labor will equal: w t = (1 αa t k α t n α t w t n t = (1 αa t kt α n 1 α t = (1 αy t w t n t = (1 α y t The BLS collects data on average labor share of income (i.e. total wage payments divided by total income. Total labor s share is roughly around two-thirds of total income payments. Hence a value of α = 0.33 is consistent with the data. In the steady state of the decentralized version of this economy, the gross real interest rate will equal the inverse of the subjective discount factor: 1 + r = 1 β. We can get data on the real interest rate from the Fisher relationship using data on nominal interest rates and inflation: r i π. Using data on nominal rates and inflation, we would say that the average real interest rate is between 2-5 percent at an annual frequency. This means that β should be between 0.95 and 0.98 at an annual frequency, or between 0.99 and if we are considering a quarterly calibration of the model. We can pin down a value of δ using data on investment as a share of output. In steady state, investment must equal δ times the capital stock: I = δ 2

3 Divide both sides by y and use the production function to get: ( I y = αa δk 1 α = δ 1 β (1 δ After normalizing a to one, this is one equation in one unknown (δ, once we have data on the investment/output ratio. Let s solve this for δ, treating I I I ( 1 β = δ ( 1 β 1 (1 δ δ = = δα (α I ( I 1 β 1 α I as a given: Private fixed investment as a share of real GDP is about percent on average in the postwar period, depending on exactly how one measures investment. If we take private fixed investment plus purchases of new durable goods, this implies that the average investment share is 24 percent. Given the calibrations of the other parameters, this implies a value of δ = at a quarterly frequency, which I will round up to δ = The calibration of the time period matters here if you are talking about annual data, for example, your β = 0.96 and implied δ would be more like Parameterizing σ, ρ, and the standard deviation of the technology shock are less straightforward. There are ways of trying to measure σ in the data, but σ doesn t show up in any steady state relationships, so from a calibration perspective it is completely free. Let s just set it at 1 so that we have log preferences. We could construct an empirical measure of TFP using growth accounting techniques to get an actual time series of ã t, which would then allows us to directly estimate ρ and the standard deviation of the shock. That would require a somewhat more complicated model; for now let s just take that these numbers to be ρ = 0.9 and σ ε = With the model solved and parameters chosen, one can engage in several different quantitative exercises. The first is to construct impulse responses to exogenous shocks. The only exogenous shock here is the TFP shock. I begin an impulse response function by assuming that the economy sits in steady state (that assumption is actually innocuous here given the definition of an impulse response function combined with the linearity of the system the IRFs are independent of starting points. We typically normalize the size of the shocks to be one standard deviation. Hence, in this case, the response of TFP on impact is The capital stock is predetermined and hence cannot respond. The response of consumption is the policy function times the responses of the states. Then we just iterate this forward compute the expected value of the states in the next period using the M matrix (i.e. E t X t+j = M j X t. 3

4 Below are the impulse responses of consumption and TFP to the TFP shock. It is an interesting exercise to fool around with some of the free parameters to see how these change: 10 x TFP Consumption Another quantitative exercise is to compute a variance decomposition. This is exactly the same idea as was presented in the structural VAR part of the course under time series. It s not a very interesting exercise here because there is only one shock by definition, this one shock must explain all of the forecast error variance of the variables in the model. In a multi-shock model, however, the variance decomposition is helpful in that it tells us which shocks are the most important in explaining variation in the observed data. The final quantitative exercise is to simulate data from the model. One can do this by drawing shocks, ε t, from some distribution (say a Normal. Then we can feed those into the system and use the M and Φ matrixes to computed simulated time paths of the variables. A common thing to do is then to (i HP filter the data and (ii look at second moment properties of the data. Some second moments of interest are typically volatilities (i.e. standard deviations, autocorrelations (a measure of persistence, and correlations with output (a measure of cyclicality. We say that a model fits the data well if these simulated second moments are similar to what we observe in the actual data. An important point whenever generating data from the model, treat the data from the model the same as you treat the actual data! The data in the model are effectively already logged and have a deterministic trend removed (i.e. linearized about the re-scaled steady state with no trend growth. Hence, you want to look at logged, detrended data in the actual data, and then HP filter those data (it actually doesn t matter whether you remove the linear trend in the actual data, because the HP filter will automatically get rid of that. 3 The State Space Representation Another way to represent the solution to these kinds of models is as state space system. This representation expresses the current states as a function of lagged states and shocks, and the current controls as functions of the current states. 4

5 Begin by partitioning the M matrix above into two components: M 1, n (n + m, are the rows corresponding to control/jump variables, and M 2, m (n + m, are the rows corresponding to the state variables. We can write out the solution for the state variables as: E t x 2,t+1 = M 2 [ x 1,t x 2,t We can break this up into two components as follows: ] E t x 2,t+1 = M 2 (:, 1 : nx 1,t + M 2 (:, n + 1 : n + mx 2,t Now use the policy function to write this solely in terms of the states: E t x 2,t+1 = (M 2 (:, 1 : nφ + M 2 (:, n + 1 : n + m x 2,t A = M 2 (:, 1 : nφ + M 2 (:, n + 1 : n + m E t x 2,t+1 = Ax 2,t Let s check the matrix dimensions to verify that this works. M 2 (:, 1 : n is m n. Φ is n m. Hence M 2 (:, 1 : nφ is m m. M 2 (:, n + 1 : n + m is m m, and x 2,t is m 1. Hence this all checks out. It is sometimes helpful to write this by lagging one period. Hence it is more common to see: E t 1 x 2,t = Ax 2,t 1 By rational expectations, we can write this as: x 2,t = Ax 2,t 1 + Bε t (5 Here ε t is dimension w 1, where w m (i.e. the number of stochastic exogenous variables must be less than or equal to the total number of states. This corresponds to the structural shocks of the system and is mean 0. Hence the matrix B is dimension m w. In the case of the stochastic growth model, w = 1 and is just the exogenous productivity shock. Hence B = [0 1]. This formulation is nice, because you can write out the states as evolving independently of the controls. Given a draw of shocks, it is then easy to simulate out the time path for the states. Then you can separately simulate values of the controls given the states, since there is just a one to one mapping between states and controls. The model can be written out: 5

6 x 2,t = Ax 2,t 1 + Bε t (6 x 1,t = Φx 2,t (7 Sometimes (6 is called the state equation and (7 the observer equation. This terminology derives from the fact that typically in economics we observe choices (e.g. consumption but not necessarily the states (e.g. TFP or even the capital stock. 6