Estimation of DSGE models: Maximum Likelihood vs. Bayesian methods

Transcription

1 Estimation of DSGE models: Maximum Likelihood vs. Bayesian methods Glenn Mickelsson Department of Economics, Uppsala University February 11, 2015 Abstract Since the seminal study by Smets and Wouters (2003) it has been popular in the business cycle literature to estimate DSGE models by postulating prior distributions to the parameters and then estimating the model using maximum likelihood (the so called the Bayesian method). However, there may be situations where the researcher wants to estimate a DSGE model with full information maximum likelihood (FIML) and avoid the use of priors entirely. A key problem with FIML is that the likelihood function may be very irregular and the solution algorithm might converge to a local instead of a global maximum. A very robust algorithm is therefore needed to find the global maximum within the relevant parameter space. This paper suggests such an algorithm and uses the model and the US data from Smets and Wouters (2007) to show that it is possible to estimate a model using FIML. Several parameter estimates are significantly different from the Bayesian estimates in Smets and Wouters (2007) and the reasons behind those differences are analyzed. Keywords: Bayesian methods, Maximum likelihood, Business Cycles, Estimate DSGE models JEL classification: E37, E32, E20 Financial support from the Jan Wallander and Tom Hedelius Foundation is gratefully acknowledged. Department of Economics, Uppsala University, Box 513, SE Uppsala, Sweden, glenn.mickelsson@nek.uu.se 1

2 1 Introduction Since the seminal paper by Smets and Wouters (2003) it has been popular in the business cycle literature to use Bayesian methods when dynamic stochastic general equilibrium (DSGE) models are estimated. Typically, Bayesian methods are implemented by postulating prior distributions of the parameters and estimating the model with maximum likelihood. Reasons for adding priors are for instance that the author have prior information based on micro founded theory and/or empirical micro studies. Another reason for priors is the difficulty in estimating large structural models, since the target function contains many local optimums with tricky patterns and global solution algorithms might either be inaccurate or too time consuming to use in practice. The biggest DSGE model this far estimated with full information maximum likelihood (FIML) is the model estimated by Ireland (2004)) which has eight structural parameters. Although, there are situations when estimates of the DSGE model using FIML is desirable. Prior information may be lacking/missing. The author may want to identify the flaws in the model by finding the parameters that get weird estimates. When trying to estimate the model with FIML one encounters the following problems/difficulties. First, local search algorithms are fast but they will almost certainly get stuck in a local optimum and thereby make the author draw the wrong conclusions. Second, currently available global search algorithms are not sufficiently good for estimating medium scale DSGE models. These points will be shown later in the paper. I propose a global solution algorithm that is better at finding the global optimum than other commonly used algorithms without being too slow to be of practical use. I use the data and the model from Smets and Wouters (2007) (SW) and estimate it using FIML. For presentation of the original model I refer to the paper by SW. In section A fast and robust global algorithm, my proposed algorithm will be explained and in section Algorithm comparisons it is compared to other global convergence algorithms by minimizing known functions. Then the fixed parameters as well as the parameter space of the DSGE estimation will be presented. The results are presented in section Results with diagnostic tests and simulated distributions of the confidence intervals. Lastly in section Analysis of differences between Bayesian and FIML estimates, there will be a discussion about the differences between the FIML estimates of this paper and the Bayesian estimates in SW. I focus on the parameter estimates in SW which are outside of the 99 % confidence interval of the FIML estimation and try to understand the reasons behind the differences. There are quite many parameters outside of the 99 % confidence interval. Investment adjustment cost φ, constant relative risk aversion σ c, habit formation in consumption h, wage stickiness ξ w, capital utilization ψ, etc. 2

3 2 A fast and robust global algorithm This section describes how the algorithm works and the procedure is illustrated with a two parameter example. A general and mathematical description is available in the Appendix. The two parameters are denoted p 1, p 2 [ 10, 10], thus min = 10 and max = 10 for both p 1 and p 1. The number of parameters is denoted P and is equal to two in this example. First, ceil(1.1 P ) = 3 vectors are created 1. The three vectors are equally spread out over (min+(max min) 0.15, max (max min) 0.15). The resulting vectors are shown in matrix A. A = (1) Then two more vectors are added that are closer to the minimum and maximum values. (min + (max min) 0.05, max (max min) 0.05). Those are shown in matrix B. B = 9 9 (2) 9 9 All of the mentioned so far vectors are summarized in matrix C for convenience. 9 C = (3) The dots in Figure 1 shows the vectors in matrix C. 1 ceil means round up to the nearest integer 3

4 Figure 1: ceil(1.1 P ) + 2 = 5 vectors that cover the parameter space (Matrix C) 2.1 Step 1 - Repeated three times Around each vector in Figure 1 another vector is stochastically chosen within the circle radius ( (max min) 0.1, (max min) 0.1) from the vectors closest to origo and ( (max min) 0.05, (max min) 0.05) for the remaining vectors. This is demonstrated with vectors 1 to 5 in Figure 2 where the circles represents the maximum and minimum just mentioned. The reason for the stochastic part is that it is hard/impossible to find a pattern which works for all kinds of problems. A stochastic part is therefore needed in order make up for the huge variation in target functions and their shapes. 4

5 Figure 2: Vectors 1 to 5 are stochastically chosen within their respective stochastic space and those are the ones that will be part of the estimation Next, in Figure 3, around vectors 1 to 5, 2 P = 4 vectors are created where each vector element is replaced by its minimum and maximum parameter space value one at the time. To demonstrate this we start by doing it for vector 1, ( 8.2, 7.8). In matrix D we see vector 1 together with the four additional vectors. 8.2 D = min max (4) min max Replacing min and max with numbers in equation (4) gives us 8.2 D = (5)

6 Figure 3: 2 P vectors created around vector 1 where vector 1 s elements are replaced with the maximum and minimum value of the parameter space [ 10, 10] one at the time. Vectors 1-5 will be run through the simplex Nelder and Mead (1965) method. Roughly speaking, the idea of the Nelder and Mead (1965) method is to move the vectors according to a schedule that is supposed to approach the optimum (minimum). After every 5 P th turn four new vectors will be created in the same manner as in Figure 3 only this time it will be around the best vector (vector with the lowest value) thus far. This entire procedure continues until the absolute logarithmic change of the best value is smaller than one; see equation (6). When the algorithm has stopped it saves that vector and continues with vector 2 in Figure 2. ln X 0 ln X 1 < 1 (6) When this is done there are five vectors which are supposed to be one step closer to the optimum. All of 6

7 these five vectors are run through the Nelder and Mead (1965) method until the absolute logarithmic change of the best value is smaller than The resulting vector is saved and step 1 is repeated two more times. This will eventually give us vectors, X 1, X 2 and X Step 2 When Step 1 is finished there are three vectors in total, X 1, X 2 and X 3. These three vectors are run through the Nelder and Mead (1965) method until the absolute logarithmic change of the best value is smaller than This vector is then run through the same procedure as in Figure 3 only this time the process is restarted every 360 P th turn. The vector with the lowest value when the improvement is lower than 10 9 is the final one! 3 Algorithm comparisons This section compares the algorithm of this paper with some other algorithms that are popular in the global optimization literature. 3.1 Algorithms Two alternative algorithms are popular global optimization algorithms in Matlab. Those are the genetic algorithm (GA) by Holland (1975) which mimics the evolution of organisms akin to Charles Darwin s natural selection principle. The other one is the simulated annealing algorithm (SA) by Kirkpatrick (1984) which was used in an econometric study by Goffe et al. (1994). SA mimics the process of hardening metal where the degree is high in the beginning and gets lower and lower until the process is stopped. Additionally there is another branch of global algorithms which build on the idea of a swarm, such as an ant colony or a bee swarm. This branch is called particle swarm optimization (PSO). Among those there is one interesting algorithm, namely the artificial bee colony (ABC) algorithm by Karaboga and Basturk (2007) which will also be used for comparison. 3.2 Objective functions To test the algorithms we need some complicated functions where the true minimum is known. Below, some functional forms, their optimum values and their bounds are presented. Additionally, a two dimensional plot is shown in order to get a visual idea of the difficulty of the optimization problem. The first five benchmark functions of the comparison are the same as those used by Karaboga and Basturk (2007). The number of parameters in this comparison are the same as in SWs model, P = 36, where P is the total number of parameters to be estimated. The first function is the Griewank function: 7

8 ( f 1 ( p) = P p 2 i i=1 ) P i=1 f 1 ( 0) = 0 p [ 600, 600] cos ( pi i ) + 1 (7) Figure 4: A plot of the Griewank function in two dimensions The Griewank function looks like a Sphere function on the broad perspective but when zooming in it becomes evident that it consists of many small local optimums. 8

9 Figure 5: A zoomed in plot of the Griewank function in two dimensions The next function is the Rastrigin function where there instead of small local minimums are bigger local minimums which are visible already on first sight: f 2 ( p) = P (p 2 i 10 cos(2πp i) + 10) i=1 f 2 ( 0) = 0 p [ 15, 15] (8) 9

10 Figure 6: A plot of the Rastrigin function in two dimensions Next we have the Rosenbrock function which looks simpler than the former ones but it has a big flat area around its global minimum which makes it harder for the algorithms to converge: f 3 ( x) = P 100(p 2 i p i+1) 2 + (1 p i ) 2 i=1 f 3 ( 1) = 0 p [ 15, 15] (9) 10

11 Figure 7: A plot of the Rosenbrock function in two dimensions Ackley is the 4th function: f 4 ( p) = 20 + e 20e ( P ) P p 2 i i=1 1 P e f 4 ( 0) = 0 p [ , ] P cos(2πp i) i=1 (10) 11

12 Figure 8: A plot of the Ackley function in two dimensions Schwefel is the 5th function and now we have a much harder function with no clear bowl-shaped pattern where it is even hard to see the minimum in a two dimensional plot: f 5 ( p) = P P p i sin p i i=1 f 5 ( , P ) = 0 p [ 500, 500] (11) 12

13 Figure 9: A plot of the Schwefel function in two dimensions The 6th function, the eggholder function, is also hard to optimize since it has many local minimums and no clear bowl shaped pattern. The basic eggholder function is two-dimensional but since all the tests will be based on a problem with the same dimensions as when we estimate a medium sized DSGE model by maxmimum likelihood estimation we must have 36 dimensions. To accomplish this the two dimensional eggholder is multiplied by itself until the number of dimensions is equal to 36. Its functional form is displayed in equation (12) ( g(p i, p i+1 ) = (p i ) sin p i+1 + pi ) ( pi ) p i sin (p i ) f 6 ( p) = P 1 ( g(p i, p i+1 )) i=1 f 6 (512, P ) = 1 p [ 512, 512] (12) 13

14 Figure 10: A plot of the Eggholder function in two dimensions Last but not least. In order to construct a very challenging problem for the algorithms, a multiplicative combination of the Eggholder and the Griewank function is created according to equation (13): f 7 ( p) = f 1 ( p)f 6 ( p) f 7 (512, , 0, 0...P ) = 1 p [ 512, 512, 600, 600] (13) 3.3 Results The comparison is carried out with 36 parameters for each function. Each function in turn was estimated ten times and the average value is presented in Table 1. The computing time in Table 2 is the total computing time for the ten repetitions. 14

15 Function SA GA ABC New algorithm Solution f 1 ( x) = Griewank e f 2 ( x) = Rastrigin e f 3 ( x) = Rosenbrock f 4 ( x) = Ackley e f 5 ( x) = Schwefel e 09 0 f 6 ( x) = Eggholder 8.413e e e f 7 ( x) = Eggholder, Griewank 3.985e e e Table 1: Functional values, the lower the better. SA=Simulated Annealing, GA=Genetic Algorithm Function SA GA ABC New algorithm f 1 ( x) = Griewank f 2 ( x) = Rastrigin f 3 ( x) = Rosenbrock f 4 ( x) = Ackley f 5 ( x) = Schwefel f 6 ( x) = Eggholder f 7 ( x) = Eggholder, Griewank Table 2: Total computing time in minutes, the lower the better. SA=Simulated Annealing, GA=Genetic Algorithm All algorithms do a decent job for the simpler functions (functions 1 to 5) except for SA which gets for the Rastrigin function. For the more complicated Eggholder and Eggholder Griewink functions only the new algorithm manages to get a reasonable values (below 14) while the others get values over Estimation of the DSGE model 4.1 Fixed parameters Following SW there are five parameters that are kept fixed. Capital depreciation, δ = 0.025, and the government spending to GDP ratio g y = 0.18 are difficult to estimate since the investment and exogenous spending ratios are not part of the estimation equation. The steady state mark up for wage setters λ w is not identified and set to 1.5. Lastly the Kimball (1995) aggregators in the goods and labor market, ɛ p and ɛ w, are both set to 10. The fixed parameters are summarized in Table 3 15

16 Description Parameter Value Depreciation δ Government spending-gdp ratio g y 0.18 Labor markup λ w 1.5 Kimball aggregator goods market ɛ p 10 Kimball aggregator labor market ɛ w 10 Table 3: Common parameters of the model. Not estimated 4.2 Intervals for estimated parameters Quantitatively the algorithm can not handle infinite numbers and we therefore have to decide upon a closed numerical set for each parameter. Habit persistence h, the calvo probabilities ξ w and ξ p, wage and price indexation ι w and ι p, capital utilization ψ, interest rate smoothing in the Taylor rule ρ and the capital share in production α are all theoretically bounded between 0 and 1. For the persistances in the autoregressive shock processes ρ a, ρ b, ρ g, ρ I, ρ r, ρ p, ρ w and ρ ga they have to be stationary and therefore belong to the interval [-0.999,0.999]. Regarding the remaining parameters such as the standard deviation of the shock processes, σ a to σ w, investment adjustment cost φ, the relative risk aversion σ c etc. it is not obvious what interval those parameters should belong to. The strategy chosen here is to specify sufficiently broad intervals and then check how quickly the likelihood function worsens as the parameter in question approaches the borders of the interval. As is evident from Figures 11 to 16, section Diagnostics, the parameter intervals in Table 4 are more than enough. 16

17 Description Parameter Interval Investment adjustment cost φ [0.01, ] Constant relative risk aversion σ c [0.25, 20.00] Habit persistence h [0.00, 1.00] Calvo parameter labor market ξ w [0.00, 1.00] Inverse Frisch elasticity σ l [0.01, 20.00] Calvo parameter goods market ξ p [0.00, 1.00] Indexation labor market ι w [0.00, 1.00] Indexation goods market ι p [0.00, 1.00] Capital utilization ψ [0.00, 1.00] Fixed production cost Φ [0.00, 30.00] T.R reaction to inflation r π [1.01, ] T.R interest rate smoothing ρ [0.00, 1.00] T.R reaction to output r y [0.00, 30.00] T.R reaction to the change in output r y [0.00, 2.00] Steady state inflation rate π [ 30.00, 30.00] Subjective discount rate times (β 1 1) [ 8.00, 4.00] Steady state hours worked l [ 30.00, 30.00] Steady state growth rate γ [0.00, 2.00] Capital share in production α [0.01, 0.99] Technology shock σ a [0.00, 5.00] Risk premium shock σ b [0.00, 5.00] Government expenditure shock σ g [0.00, 5.00] Investment shock σ l [0.00, 5.00] Monetary policy shock σ r [0.00, 5.00] Price shock σ p [0.00, 5.00] Wage shock σ w [0.00, 5.00] AR term technology shock ρ a [ 1.00, 1.00] AR term risk premium shock ρ b [ 1.00, 1.00] AR term gov. expenditure shock ρ g [ 1.00, 1.00] AR term investment shock ρ l [ 1.00, 1.00] AR term monetary policy shock ρ r [ 1.00, 1.00] AR term price shock ρ p [ 1.00, 1.00] AR term wage shock ρ w [ 1.00, 1.00] MA term price shock µ p [ 2.00, 2.00] MA term wage shock µ w [ 2.00, 2.00] Correlation between a and g ρ ga [ 1.00, 1.00] Value of the target function 17 Ω [0.00, Inf] Table 4: Parameter intervals

18 4.3 Data The data are the same as in Smets and Wouters (2007), that is, US quarterly data between 1947 to The observables are, the logarithmic change in total real GDP ( y t ), the logarithmic change in total real consumption ( c t ), the logarithmic change in total real investment ( i t ), the logarithmic change in the wage rate ( w t ), deviations of hours worked from an average work week ( l t ), the logarithmic change in the GDP deflator ( π t ) and the federal funds rate (r t ). As in Smets and Wouters (2007) the first 71 observations are excluded which results in a sample of 156 observations. 5 Results This section starts with a diagnostic test in order to verify that the estimates are indeed at the optimum. This is carried out by increasing and decreasing the value of one parameter at the time while the remaining parameters are fixed at the FIML estimate. After the Diagnostics section the simulated distributions that the confidence intervals are calculated from will be presented in the Simulated distributions section. Lastly, the key part of this paper will be presented in FIML versus Bayesian estimation - Estimates where the FIML estimates and their confidence intervals are presented together with the corresponding values in SW. 5.1 Diagnostics The FIML estimates in Table 5 are evaluated by decreasing and increasing one parameter at the time while the remaining parameters are kept constant at the FIML estimate. If a minimum has in fact been reached the resulting plot should have its lowest point at the estimate (solid vertical line) and the objective function should increase as the parameter moves away from that value in either direction. This is the case when looking at Figures 11 to 16 where the vertical solid line is the estimate. The dashed vertical lines show SW s mode estimates. These diagnostic plots are also interesting since they indicate whether or not the initial intervals of the estimation in Table 4 are too small or not. According to the diagnostics, virtually all intervals have more than enough margin so that the minimum is an interior point in the interval. The exceptions are parameters that are either theoretically bounded or bounded for stationarity reasons. Worth mentioning here are the indices, ι w, ι p and capital utilization, ψ, as well as the persistence for some of the AR(1) processes, ρ a, ρ g, ρ I and ρ w. Those seem to be at or near a corner solutions and this issue is discussed in Analysis of differences between Bayesian and FIML estimates. Except for those, all parameters seem to be at an optimum where the target function clearly worsens as the parameter value moves away from its FIML estimate. 18

19 Figure 11: Diagnostics where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 19

20 Figure 12: Diagnostics where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 20

21 Figure 13: Diagnostics where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 21

22 Figure 14: Diagnostics where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 22

23 Figure 15: Diagnostics where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 23

24 Figure 16: Diagnostics where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 5.2 Simulated distributions In Smets and Wouters (2007) it is assumed that the shocks have the properties in where ε t N (0, Σ) (14) ε t = ( ε a t, ε b t, ε g t, ε I t, ε r t, ε p t, ε w ) t (15) and where the diagonal elements of Σ consists of the variances in Table 6 and the off diagonal elements are zero. 24

25 Thus, all stochastic shocks follow a normal distribution with zero mean and they are identically and independently distributed. Based on SW s assumption regarding Σ and the estimates in Table 5 and Table 6 column FIML new samples are simulated with the same sample size (230) as the original data. Then the optimal vector is estimated for each sample. This procedure is repeated until the distributions in Figure 17 to Figure 21 have converged; For convergence criteria see Convergence diagnostics of the histograms in the Appendix. To find new estimates I used the FIML estimates as the guessed input vector and only the very last part of the solution algorithm was performed. Three things are important. 1. That the estimated parameter vector generates a larger likelihood than the original vector. 2. The distributions have converged such that it does not affect the confidence levels. 3. The converged distributions should look like an approximative mirror image of Figures 11 to 16. Roughly speaking, this is also the case when comparing the figures in the Diagnostics section with the figures in the Simulated distributions sections. 25

26 Figure 17: Simulated distributions where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007). The number of simulated samples=

27 Figure 18: Simulated distributions where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007). The number of simulated samples=

28 Figure 19: Simulated distributions where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007). The number of simulated samples=

29 Figure 20: Simulated distributions where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007). The number of simulated samples=

30 Figure 21: Simulated distributions where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007). The number of simulated samples= FIML versus Bayesian estimation - Estimates Tables 5 and 6 show the most important tables of the paper, namely the mode estimate, its confidence interval and a comparison with SWs posterior estimates. *, ** and *** indicates if SWs mode estimates are outside of this papers 90%, 95% and 99% confidence interval respectively. 30

31 Description Parameter 95% FIML 95% SW FIML SW Investment adjustment cost φ [0.56, 2.05] [3.97, 7.42] *** Constant relative risk aversion σ c [2.11, 6.39] [1.16, 1.59] *** Habit persistence h [0.10, 0.29] [0.64, 0.78] *** Calvo parameter labor market ξ w [0.84, 0.97] [0.60, 0.81] *** Inverse Frisch elasticity σ l [0.60, 3.43] [0.91, 2.78] Calvo parameter goods market ξ p [0.61, 0.81] [0.56, 0.74] Indexation labor market ι w [0.71, 0.99] [0.38, 0.78] ** Indexation goods market ι p [0.00, 0.20] [0.10, 0.38] ** Capital utilization ψ [0.97, 0.99] [0.36, 0.72] *** Fixed production cost Φ [1.10, 1.52] [1.48, 1.73] *** T.R reaction to inflation r π [3.21, 22.77] [1.74, 2.33] *** T.R interest rate smoothing ρ [0.95, 1.00] [0.77, 0.85] *** T.R reaction to output r y [0.14, 2.08] [0.05, 0.12] *** T.R reaction to the change in output r y [0.28, 0.44] [0.18, 0.27] *** Steady state inflation rate π [0.24, 1.40] [0.61, 0.96] Subjective discount rate times (β 1 1) [ 2.22, 0.00] [0.07, 0.26] ** Steady state hours worked l [ 3.44, 6.77] [ 1.30, 2.32] Steady state growth rate γ [0.36, 0.47] [0.40, 0.45] Capital share in production α [0.10, 0.18] [0.16, 0.21] ** Table 5: FIML=Full information maximum likelihood mode estimates. SW=The Bayesian posterior estimates in Smets and Wouters (2007). SW s estimates outside of the FIML confidence intervals are denoted *=90%, **=95% and ***=99% respectively. 31

32 Description Parameter 95% FIML 95% SW FIML SW Technology shock σ a [0.43, 0.60] [0.41, 0.50] Risk premium shock σ b [0.05, 0.12] [0.19, 0.27] *** Government expenditure shock σ g [0.46, 0.57] [0.48, 0.58] Investment shock σ l [0.46, 2.27] [0.37, 0.53] *** Monetary policy shock σ r [0.21, 0.27] [0.22, 0.27] Price shock σ p [0.09, 0.16] [0.11, 0.16] Wage shock σ w [0.26, 0.33] [0.20, 0.28] *** AR term technology shock ρ a [0.97, 1.00] [0.94, 0.97] *** AR term risk premium shock ρ b [0.63, 0.92] [0.07, 0.36] *** AR term gov. expenditure shock ρ g [0.98, 1.00] [0.96, 0.99] ** AR term investment shock ρ l [0.92, 0.99] [0.61, 0.80] *** AR term monetary policy shock ρ r [ 0.35, 0.05] [0.04, 0.24] *** AR term price shock ρ p [0.86, 0.99] [0.80, 0.96] AR term wage shock ρ w [0.88, 0.99] [0.94, 0.99] MA term price shock µ p [0.60, 0.98] [0.54, 0.85] MA term wage shock µ w [0.88, 0.99] [0.75, 0.93] * Correlation between a and g ρ ga [0.38, 0.70] [0.37, 0.66] Value of the target function Ω [ , ] [ Inf, 0.00] Table 6: FIML=Full information maximum likelihood mode estimates. SW=The Bayesian posterior estimates in Smets and Wouters (2007). SW s estimates outside of the FIML confidence intervals are denoted *=90%, **=95% and ***=99% respectively. 6 Analysis of differences between Bayesian and FIML estimates In this section we look more closely at the parameters in Tables 5 and 6 where SW s estiamtes are outside of the 99 % confidence intervals from the FIML estimation. For each parameter we first examine what role it plays in the model and then we try to understand why the FIML and SW s estimates are different. FIML and SW s estimate are denoted p F IML and p SW respectively for parameter p. 6.1 Investment adjustment cost φ Households own the capital and log linearizing the household s optimal condition for capital holdings gives i t = 1 (1 + βγ (1 σc) )γ 2 φ q 1 t βγ i (1 σc) t 1 + βγ(1 σc) 1 + βγ E ti (1 σc) t+1 + ε i t (16) where q t is the real value of the existing capital stock, defined as 32

33 1 δ q t = R k + 1 δ E R k tq t+1 + R k + 1 δ E trt+1 k (r t E t π t+1 ) ε b t (17) q t is positively correlated with a weighted average of the expected real value of capital, E t q t+1, and the expected future rental rate of capital, E t r k t+1. These weights are 1 δ R and R k +1 δ k R k +1 δ respectively, where Rk is the steady state rental rate of capital and ε b t is a shock to the risk premium of investment. As is evident from equation (16), if the expected future real value of capital increases the household will save more in capital at a decreasing rate determined by φ. The reason for this is that a higher φ means that a bigger share of investment is wasted due to the adjustment cost. The FIML estimates suggests a lower investment cost where φ F IML = 1.28 compared to φ SW = 5.48 i t 1 = q t (1 + βγ (1 σc) )γ 2 φ Equation (18) shows how many percentages investment goes up if the value of real capital goes up by one percent, ceteris paribus. The answer to that question is 0.39 percent for the FIML estimate and 0.09 percent for SW. That SW get a much lower value is not surprising since they have the prior φ N(4, 1.5) which has a center way above the FIML estimate. SW base their prior choice on estimates from Christiano et al. (2005) whose investment model is a bit simpler with fewer parameters and where investment depends on the real rental price of capital instead of the real value of capital. The analogous equation in Christiano et al. (2005) for investment is i t = i t φ (18) β j E t 1 rt+j k (19) j=0 They estimate φ by matching empirical impulse responses of their model and get the estimate φ = 2.48 which gives an analogous of 0.4 percent to equation (18). From this we can conclude that SW used a little higher prior value of φ than justified by the estimate in Christiano et al. (2005). However, neither the real rental price of capital, r k t, nor the real value of capital, q t, can be directly observed in the data which is why conclusions based on those variables alone might not be enough. Simply speaking, an important reason for adding adjustment costs in investment is to make the model better match empirical fluctuations in investment. The standard deviation of the change in investment, σ i, divided by the standard deviation of the change in output, σ y, is an interesting measure of investment fluctuations since it is decreasing in φ. In the data, σ i σ y = When simulating the DSGE model using the FIML and SW estimates we get 2.62 and 2.13 respectively. This strengthens the previous findings, namely that SW have a high prior value of φ. 33

34 6.2 The subjective discount factor β, relative risk aversion σ c and habit persistence in consumption h In order to reduce the income effect on labor supply SW use a utility function where consumption and labor are complements when σ c > 1, this makes the income effect in labor supply independent of σ c ; for a discussion of the role of utility functions see Basu and Kimball (2002). Households optimize the utility stream in (20) with respect to consumption C j,t, hours worked L j,t, investment I j,t, bond holdings B j,t and capital utilization Z j,t. E t s=0 subject to the budget constraint ( β s Cj,t+s h C ) 1 σc ( ) j,t+s 1 σc 1 exp L 1+σc j,t+s 1 σ c 1 + σ l (20) B j,t+s C j,t+s +I j,t+s + ε b T t+s B j,t+s 1 tr t+s P t+s and the capital accumulation equation P t+s + W h j,t+s L j,t+s P t+s K j,t = (1 δ) + ε i t + Rk t+sz j,t+s K j,t+s 1 a (Z j,t+s ) K j,t+s 1 + Div t+s P t+s P t+s (21) [ 1 S I ] j,t I j,t (22) I j,t 1 The subjective discount factor β decides how the household values the future relative to today. The estimated parameter that indirectly decides β is defined as Solving for β x = 100(β 1 1) (23) β = 100 x The estimates of x implies β SW = and β F IML = respectively. To investigate the plausibility of the estimates let us look at the household s Euler equation in steady state. (24) β = γσc Π r (25) where γ is the steady state growth rate in consumption, Π is the steady state growth rate in inflation and r is the steady state interest rate. Using averages from the data together with the estimates of σ c gives β SW = and β F IML = respectively. This implies that both estimates make sense. Although, it is important to remember that by restricting β σ c is indirectly restricted. This is seen when solving for σ c in equation (25) 34

35 σ c = ln ( ) 100+r Π100 ln ( 1 + γ ) (26) 100 SW restrict β to be between 0.98 and 1 despite that equation (25) clearly demonstrates that having β > 1 is possible in a model with consumption growth and inflation growth. SW assumption restricts σ c to be between -3 and as the lower bound is not too high since negative values of σ c is not theoretically possible. Restricting β to be at most 1.48 on the other hand might be a bit low which will be discussed this later. The log-linearized Euler equation is c t = h γ [ h γ c t 1 + E t c t+1 + σ c 1 σ c W h L C (l t E t l t+1 ) 1 h γ σ c (r t E t π t+1 + ε b t) Let us first discuss how c t relates to lagged consumption, c t 1. The dependens that c t 1 has on c t is mostly decided by the habit parameter h. More specifically the coefficient relating c t to c t 1 is ] (27) ɛ c dc t = h/γ dc t h/γ (28) SW have a much higher habit formation, h SW = 0.71 compared to h F IML = 0.2, so that ɛ c,f IML = and ɛ c,sw = respectively. A key parameter that decides the elasticity of hours worked, l t, with respect to current consumption c t is the risk aversion parameter σ c. The higher the σ c, given that σ c > 1, the more are consumption and hours worked complements which is evident in equation (29) ɛ c,l dc t = σ c 1 dl t σ c W h L C (29) Since σ c,f IML = 4.24 the FIML estimate suggests a stronger complementary effect and when plugging in numbers we get ɛ c,l,f IML = and ɛ SW,c,l = Additionally, σ c affects the households sensitivity to interest rate changes together with the habit parameter h. Both σ c and h weaken this sensitivity which is evident in equation (30) ɛ c,r dc t = γ/h 1 1 (30) dr t γ/h + 1 σ c ɛ c,r is decreasing in both σ c and h which means that a high h and a low σ c can generate a similar interest rate sensitivity as a low h and a high σ c. This is the reason why the FIML estimate and the SW estimate are similar; ɛ c,r,f IML = and ɛ c,r,sw = respectively. Thus, if the risk adjusted real interest rate goes up by 1 percent, current consumption goes down by about 0.14 percent for both estimates. SW? Why does the FIML estimates suggest a lower habit formation and a higher risk aversion compared to 35

36 Referring to the discussion about β in the beginning of this section. When β is restricted to be at most 1, σ c is implicitly restricted to be at most Using the value for ɛ c,r, which is around -0.14, implicitly restricts h to be at least 0.66; see equation (31). h = γ 1 + σ cɛ c,r 1 σ c ɛ c,r (31) Furthermore, SW s priors for risk aversion and habit formation are σ c N (1.4, 0.37) and h β (0.7, 0.1) respectively which also steers the results towards a high habit and low risk aversion. SW justify their prior choice of h with the fact that the data suggests a high autocorrelation in consumption and to habit formation studies such as Fuhrer (2000) and McCallum and Nelson (1999). 6.3 Wage stickiness (ξ w ) and labor and goods market indexation (ι w and ι p ) Unions set wages according to a sticky price mechanism a la Calvo (1983). The wage rate W t+s depends on W t (l), the wage rate of members belonging to union l that are allowed to change their wage in period t and an index. The index is a weighted sum of past price inflation Π s i=1 γπιw t+i 1 and steady state price inflation π according to (32) W t+s (l) = W t (l)π s i=1γπ ιw t+i 1 π1 ιw (32) The FIML estimate ι w = 1 means that past price inflation gets all the weight and steady state inflation none. The objective of the union, allowed to change its wage, is to decide an optimum wage, Wt (l), that will last until they are allowed to change it again. The probability of change each period is 1 ξ w and is known to the unions. The duration of the wage contract is thus expected to last for 1 1 ξ w periods. ι p has the analogous purpose as ι w when firms set their prices according to a Calvo (1983) mechanism where the price evolves according to. P t+s = P t (i)π s j=1γπ ιp t+j 1 π1 ιp (33) The FIML estimate ι p = 0 means that past inflation plays no role and the economy ends up with a regular forward looking Phillips curve; see for instance Gali (2009). Now let us turn to the most important parameter for the dynamics, namely wage stickiness ξ w. The estimates are ξ w,sw = 0.74 and ξ w,f IML = 0.9 which corresponds to a wage contract duration of 3.8 and 10 quarters respectively. The estimate by SW is significantly lower than the FIML estimate, so let us investigate why this is the case. SW use the prior ξ w β (0.5, 0.1). They assume that price stickiness is similar to wage stickiness and base their prior on the micro evidence on price stickiness from Bils and Klenow (2002). There are not that many studies about wage stickiness but one is Barattieri et al. (2010) where they look at wage surveys and find that ξ w is between 0.82 and 0.95 which corresponds to a duration between 5.6 and 20 quarters. This 36

37 is in line with the FIML estimate and suggests that SWs estimate, ξ w = 0.74, is too low because of an inappropriately chosen prior. Higher wage stickiness implies a much slower pass through of the marginal rate of substitution into wages which in turn decreases the variance of the wage. This is the most dominant effect of the wage dynamics. The indexation parameter ι w has only a small effect in comparison. 6.4 Capital utilization ψ Capital utilization ψ makes it possible to change production in the short run without hiring/firing nor changing the capital stock. The role of capital utilization can be explained by the log linearized production function y t = φ p (αk s t + (1 α)l t + ε a t ) (34) where k s t is the capital stock that is actually used in production and can be temporary increased/decreased when the household orders a higher/lower utilization rate of current installed capital k t 1. The possibility to increase the capital stock in excess of installed capital, kt s k t 1, is given by ( ) 1 kt s k t 1 = ψ 1 rt k (35) The utilization rate is positively correlated to the real rental rate of capital r k t since a higher r k t makes it more profitable to increase utilization. ψ determines the cost of utilizing capital and is normalized to be between 0 and 1. If ψ = 1 capital is infinitely expensive to utilize and the utilization channel is completely shut down so that k s t = k t 1. This is what the FIML estimation suggests since ψ F IML = 1. On the contrary, SW have capital utilization as an important feature with ψ SW = SW s estimate is significantly lower than the FIML estimate, so let us figure out why this is the case. A low ψ dampens the effect that output has on the real rental rate of capital which in turn dampens the effect in marginal cost. This is the main reason why Christiano et al. (2005) introduced this particular utilization mechanism on capital in the first place. How can we explain the difference between the FIML and SW results? Since wage stickiness is higher for the FIML estimate, ξ w,f IML = 0.9, compared to ξ w,sw = Wage pass through is low which makes the reaction in marginal cost low as well despite a big effect of production on the rental rate of capital. This effect can be seen in equation (36) mc t = αr k t + (1 α)w t (36) The reason why changes in wages have a dominant influence on marginal cost is because the income share of capital is much smaller than the income share of labor, α F IML = 0.14 and α SW = Since SW use the sticky wage prior ξ w β (0.5, 0.1) wages respond relatively strongly and the model is therefore forced to utilize capital in order to dampen the total effect of production on marginal cost. Additionally, SW use the prior ψ β (0.5, 0.15) for capital utilization which also steers the estimate towards more variation in capital utilization and a lower wage stickiness. 37

38 6.5 Fixed production cost Φ The fixed production cost enters in the production function y t = Z t k α t 1L (1 α) t Φ (37) Rewriting the production function in terms of logarithmic deviations from steady state gives ŷ t = ( 1 + Φ ) ( ) y αˆk t s + (1 α)ˆl t + ε a t (38) where ŷ t = y t y and means the variable in steady state. The reason why ŷ t is increasing in Φ is because a higher fixed production cost decreases y t which in turn increases the dynamic response measured in percent. This is demonstrated in Figure 22 below when subjecting the economy to the risk shock ε b t. 38

39 Figure 22: Impulse response of a the risk shock ε b t where the black solid line is the FIML estimate and the red dashed line is the mode estimate from Smets and Wouters (2007) 6.6 Taylor rule parameters r π, ρ, r y and r y The FIML estimates are much higher than SW s estimates for all the Taylor rule parameters. The central bank reacts strongly to inflation and output deviations. Still, the total reaction is weaker compared to SW due to much higher interest rate smoothing, ρ F IML = 0.98 and ρ SW = The stochastic processes The FIML estimates have larger shock variances and a few of the AR(1) processes get really high persistences where ρ a,f IML, ρ g,f IML, ρ I,F IML and ρ w,f IML get values that are close to one. This indicates that the assumptions regarding the shocks might be too restrictive. SW assume that all shock persistences follow a 39

40 beta distribution with mean 0.5 and standard deviation 0.2. This pushes down the results from one and hides the fact that the model needs other assumptions regarding the shock processes. To highlight the problem let us look at it in terms of a simple production function and a shock to technology respectively. Y t = K α t (A t L t ) 1 α (39) A t = e γ A ρa t 1 eεa t (40) where ρ a ( 1, 1), ε a t N(0, σ 2 a) and e γ is a deterministic trend. If for instance the true value is ρ a = 1.2 equation (40) will get explosive dynamics which forces the FIML estimate to chose the closest possible value which is ρ a,f IML = Below follows a potential solution to this problem. First, rewrite the model in differences and ratios only such that where K t = Kt Y t and K t = Kt Y t. Now the production function becomes K t = K t Y t Kt 1 (41) Y t = K α t ( A t L t ) 1 α (42) Furthermore, we also need to make different assumptions regarding the shock process A t = e γ X ρx t (43) where X t = X ρx t 1 eεx t (44) and ρ x ( 1, 1) ε x t N(0, σ 2 x) (45) Now we can estimate ρ x and σ 2 x even in cases where ρ a > 1. This is a suggestion for further research, rewrite the model in SW using the suggestion above, estimate it using the same methodology as is in this paper and see if the model parameters get significantly different values. 6.8 Conclusion - Differences Priors are not innocuous and affects the results quite a lot even in cases where there are better and more defined information in the data. The importance that each shock has on the variables are also quite different and evident when looking at the variance decomposition in Table 7. 40

41 Variable A Risk G I MP Cost Wage Y (0.44, 0.17) (0.28, 0.26) (0.17, 0.37) (0.01, 0.13) (0.09, 0.05) (0.01, 0.02) (0.00, 0.00) C (0.22, 0.02) (0.42, 0.82) (0.12, 0.01) (0.07, 0.00) (0.14, 0.11) (0.02, 0.01) (0.02, 0.02) I (0.05, 0.03) (0.31, 0.04) (0.00, 0.00) (0.54, 0.88) (0.10, 0.03) (0.00, 0.02) (0.01, 0.00) W (0.06, 0.02) (0.00, 0.01) (0.00, 0.00) (0.00, 0.00) (0.00, 0.00) (0.24, 0.29) (0.70, 0.67) N (0.06, 0.23) (0.47, 0.24) (0.29, 0.34) (0.01, 0.12) (0.15, 0.05) (0.01, 0.01) (0.00, 0.01) Π (0.11, 0.04) (0.00, 0.00) (0.00, 0.00) (0.00, 0.01) (0.01, 0.02) (0.71, 0.72) (0.17, 0.20) Table 7: Variance decomposition table for the shocks. The left number in brackets have used the FIML estimates and the right number have used the SW estimates. A=Technology shock, Risk=Risk premium shock, G=Government expenditure shock, I=Investment shock, MP= Monetary policy shock, Cost=Cost push shock, Wage=Wage push shock. The persistence of the technology shock is probably why this shock has a much bigger variance share on both output and consumption. 41

42 7 Conclusion In this paper I suggest an algorithm that makes it possible to estimate a medium sized DSGE model using full information maximum likelihood (FIML) without the need for priors (The Bayesian method). The method is applied to the model by Smets and Wouters (2007) (SW). Simulated distributions are used to make inference. FIML allows the researcher to easier find flaws in the model and generates estimates even in cases where prior information is missing or poor. Quite many of the FIML estimates are significantly different from those of SW at the 99 % level. An important difference is that the FIML estimate suggests a stickier wage than SW. SW s estimate is based on a poor prior since it comes from micro data on price stickiness. A consequence of a stickier wage is smaller variance in marginal cost which in turn makes the need to decrease the variance in the real rental price of capital unnecessary. For this reason the FIML estimate shuts down the capital utilization mechanism entirely. Furthermore, the FIML estimate gets very persistent shock processes to technology, government expenditure, investment and the wage Phillips curve. This indicates that the stationarity restrictions, steaming from SW s way of modeling the economy, generates the wrong estimates for some of the shock persistences. Modeling in differences might solve this problem which is demonstrated with a short example. The FIML estimates suggests low habit formation in consumption and high risk aversion while SW s estimates suggest the opposite. The reason why SW get their result is because they use priors on those parameters and on the subjective discount rate β. It is not easy to say which values are more correct in this case, since different combinations of habit and risk aversion can generate similar results. The interest rate sensitivity of consumption is similar in both FIML and SW. SW implicitly restricts risk aversion in consumption σ c to a low value, by restricting the subjective discount factor β to be at most one. But in a model where there is growth in inflation and consumption it is quite possible for β to be larger than one. References Barattieri, Alessandro, Susanto Basu, and Peter Gottschalk (2010), Some evidence on the importance of sticky wages. Working Paper 16130, National Bureau of Economic Research, URL org/papers/w Basu, Susanto and Miles Kimball (2002), Long-run labor supply and the elasticity of intertemporal substitution for consumption. manuscript, University of Michigan, URL public/readinggroup/shenghao pdf. Bils, Mark and Peter J. Klenow (2002), Some evidence on the importance of sticky prices. Working Paper 9069, National Bureau of Economic Research, URL 42

43 Calvo, Guillermo A. (1983), Staggered prices in a utility-maximizing framework. Journal of Monetary Economics, 12, , URL Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans (2005), Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy, 113, 1 45, URL http: // Fuhrer, J. C. (2000), Habit formation in consumption and its implications for monetary-policy models. American Economic Review, , URL Gali, Jordi (2009), Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework. Princeton University Press. Goffe, William L., Gary D. Ferrier, and John Rogers (1994), Global optimization of statistical functions with simulated annealing. Journal of Econometrics, 60, 65 99, URL science/article/pii/ Holland, John H. (1975), Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence, volume viii. U Michigan Press, Oxford, England. Ireland, Peter N. (2004), A method for taking models to the data. Journal of Economic Dynamics and Control, 28, , URL S Karaboga, Dervis and Bahriye Basturk (2007), A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization, 39, , URL Kimball, Miles S. (1995), The quantitative analytics of the basic neomonetarist model. Working Paper 5046, National Bureau of Economic Research, URL Kirkpatrick, Scott (1984), Optimization by simulated annealing: Quantitative studies. Journal of Statistical Physics, 34, , URL McCallum, Bennett T. and Edward Nelson (1999), Nominal income targeting in an open-economy optimizing model. Journal of Monetary Economics, 43, , URL science/article/pii/s Nelder, J. A. and R. Mead (1965), A simplex method for function minimization. The Computer Journal, 7, , URL 43

44 Smets, Frank and Raf Wouters (2003), An estimated dynamic stochastic general equilibrium model of the euro area. Journal of the European Economic Association, 1, , URL stable/ Smets, Frank and Rafael Wouters (2007), Shocks and frictions in US business cycles: A bayesian DSGE approach. The American Economic Review, 97, , URL A Algorithm description A.1 Global part Each parameter p has its min and max value, p min and p max. First let us introduce the new min and max definitions and p minc = p min + (p max p min )c (46) p maxc = p max (p max p min )c (47) where c is a small number and the idea is to have bounds that are a little bit apart from the bounds than p min and p max. The next step is to create N vectors where each vector has the dimension P x1 where P is the number of parameters. Each element of every vector is generated with the following function x i p = p minc + (p maxc p minc ) i N (48) All the resulting N vectors are stored in a matrix x 1 1 x 2 1 x N 1 x 1 2 x 2 2 x N (49) x 1 P x 2 P x N P Each element is multiplied with the stochastic process ε i p uniform(1 δ, 1 + δ) (50) where δ is a small number and the idea is to get a small stochastic variance around all parameters. 44

45 ε 1 1x 1 1 ε 2 1x 2 1 ε N 1 x N 1 ε 1 X ε = 2x 1 2 ε 2 2x 2 2 ε N 2 x N (51) ε 1 P x1 P ε 2 P x2 P ε N P xn P Additionally two more vectors are added with the first one very close to origo and the second one very close to the maximum value of all parameters. This in order to catch the minimum and maximum parts of the space as well. and X min = X max = ε min 1 ε min 2. ε min P ε max 1 ε max 2. ε max P, ε min p uniform (p min, p min + (p max p min )δ) (52), ε max p uniform (p max (p max p min )δ, p max ) (53) [ ] X = X min X max X ε P x2+n (54) Each vector in matrix 54 are evaluated the following way. First matrix A is created around vector X i x 1 x min 1 x 1 x 1 x max 1 x 1 x 1 [ ] x 2 x 2 x min A = X 2 x 2 x 2 x max 2 x 2 Amin Amax = x P x P x P x min P x P x P x max P (P,1+2P ) Matrix A is then run through the Nelder and Mead (1965) simplex method until log X < 1 where X is the vector with the best function value. Additionally, the process is restarted every 5 P th time around the vector with the best function value thus far in order not to get stuck in a local minimum. This step is repeated for each one of the vectors which result in 2 + N vectors. These vectors are then sorted in order of the best value. The P + 1 vectors that have the best value are saved in a a new matrix and are run through the Nelder and Mead (1965) simplex method. The resulting vector is then saved. This entire process is repeated three times. 45

46 A.2 Local part Three vectors are saved as a result of the global part, these three vectors are in turn run through the Nelder and Mead (1965) simplex method. This creates an output vector which in turn is used to create matrix A as in the local part and is run trough the Nelder and Mead (1965) simplex method. That resulting vector is the final vector! A.3 The Nelder and Mead (1965) algorithm - applied to this example The algorithm consists of the following steps. Step 1. Order all vectors in Matrix D (equation (4)) in the order of their target values. Calculate the center of the P-1 best vectors. From the center walk approximately in the opposite direction from the worst vector. If this reflection vector is better than the best vector investigate this direction further by going further in the same direction (extend). If the extended vector is not better than the reflection vector replace the worst vector with the reflection vector. If the extended vector is better than the reflection vector replace the worst vector with that one instead. Then repeat step 1. If the reflection vector is not better than the second worst vector but better than the worst vector make an outside contraction. That is, use the same idea as reflection but make a smaller step. Replace the worst vector if it is better than the reflection point and repeat step 1. If the outside contraction vector is not better than the reflection vector create a weighted average with the center and the worst vector. If this vector is better than the worst vector replace it and repeat step 1. If it is not better than the worst vector perform a shrink where all vectors take one step closer to the best vector. Repeat step 1. B Convergence diagnostics of the histograms The thick horizontal lines in Figure 23 to 26 below show Pmin 95% 95% (i) and Pmax (i) for each variable. Pmin 95% (i) and P 95% max (i) is the lower and upper bound of the 95 % confidence interval respectively for parameter P based on i simulations. i is the value of the x-axis and i The more simulations (higher i) the more certain should the interval become. 46

47 Figure 23: Convergence of the distributions in section Simulated distributions. The black solid horizontal line is the FIML estimate, the red dashed line the SW estimate and the blue thick horizontal lines shows Pmin 95% 95% (i) and Pmax (i) respectively. 47

48 Figure 24: Convergence of the distributions in section Simulated distributions. The black solid horizontal line is the FIML estimate, the red dashed line the SW estimate and the blue thick horizontal lines shows Pmin 95% 95% (i) and Pmax (i) respectively. 48

49 Figure 25: Convergence of the distributions in section Simulated distributions. The black solid horizontal line is the FIML estimate, the red dashed line the SW estimate and the blue thick horizontal lines shows Pmin 95% 95% (i) and Pmax (i) respectively. 49

50 Figure 26: Convergence of the distributions in section Simulated distributions. The black solid horizontal line is the FIML estimate, the red dashed line the SW estimate and the blue thick horizontal lines shows Pmin 95% 95% (i) and Pmax (i) respectively. 50