Topic 5: Stochastic Growth and Real Business Cycles

Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45

Lag Operators The lag operator (L) is de ned as Similar to the forward operator (F ). Two key properties: Lx t = x t 1. (1) L (x t + y t ) = x t 1 + y t 1, (2) L (Lx t ) = L 2 x t = x t 2, (3) L n x t = x t n. (4) White noise process is a sequence of random variables that have mean 0 and variance σ 2 and are uncorrelated: E [ε t ] = 0, E ε 2 t = σ 2, E [ε t ε k ] = 0 for t 6= k. (5) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 2 / 45

Stationarity De nition Strict stationarity: a process y t is strict stationary if the probability distribution of (y t, y t+1,, y t+k ) is identical to that of (y τ, y τ+1,, y τ+k ) for all t, τ, and k. (i.e., all joint distributions are time invariant.) De nition Weak stationarity (or covariance-stationary): a process y t is called covariance-stationary if the mean and autocovariances do not depend on time: E [y t ] = µ, 8t (6) E [(y t µ) (y t j µ)] = γ j for all t and j. (7) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 3 / 45

Moving Average Process A moving average process, MA(q), is a stochastic process that is a weighted sum of the q most recent values of its stochastic term ε t, where ε t is a white noise: Properties of MA(q): y t = µ + ε t + θ 1 ε t 1 + θ 2 ε t 2 + + θ q ε t q. (8) = mean: E [y t ] = µ + E [ε t ] + θ 1 E [ε t 1 ] + + θ q E [ε t q ] = µ variance: var [y t ] = γ 0 = E [y t µ] 2 = 1 + θ 2 1 + + θ 2 q σ 2 j-th autocovar: γ j = E [(y t µ) (y t j µ)] θj + θ j+1 θ 1 + θ j+2 θ 2 + + θ q θ q j for j = 1,, q 0 for j > q. where we use the fact that E [ε t ε k ] = 0 for t 6= k. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 4 / 45

Autoregressive (AR) Process An example of autoregressive process, AR(q), is AR(1), y t = c + ρy t 1 + ε t. (9) where ε t is a white noise. We can easily solve this stochastic di erence equation backwards: y t = c k 1 ρ j k 1 + ρ j ε t j + ρ k y t k. (10) j=0 j=0 As k goes to, the above sum is well de ned only if jρj < 1. This is the condition for an AR(1) to be covariance-stationary. These AR(1) processes are called stable. In this case, the last term goes to 0 if we keep on iterating to : y t = c 1 ρ + ρ j ε t j, (11) j=0 which is the MA( ) representation of AR(1). Note that we can also use the Lag operator to obtain this result. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 5 / 45

Properties of AR(1): mean:e [y t ] = µ = c 1 ρ. (12) variance:var [y t ] = γ 0 = E [y t µ] 2 = σ2 1 ρ 2. (13) j-th autocov:γ j = E [(y t µ) (y t j µ)] = ρj σ 2 for all j. 1 ρ2 (14) AR(2): y t = c + ρ 1 y t 1 + ρ 2 y t 2 + ε t. (15) Using the lag operator, we have 1 ρ 1 L + ρ 2 L 2 y t = c + ε t. (16) The stability of this process depends on the polynomial 1 ρ 1 z + ρ 2 z 2 = 0 (or (1 λ 1 z) (1 λ 2 z) = 0). The process is stable if both of the roots are bigger than 1 in absolute value. (If the roots are complex, both of them must lie outside the unit circle.) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 6 / 45

c Properties of AR(2): E [y t ] = µ = 1 ρ 1 ρ. To nd second moments 2 (variance and autocov), we rewrite (15) as y t µ = ρ 1 (y t 1 µ) + ρ 2 (y t 2 µ) + ε t (17) and multiply both sides of it by y t j µ. Taking expectations: γ j = ρ 1 γ j 1 + ρ 2 γ j 2, for j = 1, 2,. The autocorrelations are found by dividing it by γ 0 : ψ j = ρ 1 ψ j 1 + ρ 2 ψ j 2, for j = 1, 2,. (18) Setting j = 1 and 2, using the fact that ψ 0 = 1 and ψ 1 = ψ 1, ψ 1 = ρ 1 1 ρ 2, ψ 2 = ρ 1 ψ 1 + ρ 2. Multiplying both sides of (17) by y t µ: γ 0 = ρ 1 γ 1 + ρ 2 γ 2 + σ 2 = ρ 1 γ 0 ψ 1 + ρ 2 γ 0 ψ 2 + σ 2 ) γ 0 = 1 ρ h 2 i σ 2. (19) (1 + ρ 2 ) (1 ρ 2 ) 2 ρ 2 1 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 7 / 45

More General Case AR(p): Properties of AR(p): y t = c + ρ 1 y t 1 + ρ 2 y t 2 + + ρ p y t p + ε t. (20) mean: E [y t ] = µ = c 1 ρ 1 ρ 2 ρ p. variance: var [y t ] = γ 0 = ρ 1 γ 1 + ρ 2 γ 2 + + ρ p γ p + σ 2. j-th autocov: γ j = ρ 1 γ j 1 + ρ 2 γ j 2 + + ρ p γ j p. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 8 / 45

Unconditional and Conditional Expectations Again For the AR(1) process, unconditional and conditional expectations are di erent: E [y t ] = µ = c 1 ρ while E t [y t+1 ] = c + ρy t 6= c 1 ρ. (21) Law of iterated expectation: E t [E t+1 [y t+2 ]] = E t [y t+2 ], (22) which means that what you expect to expect tomorrow about the day after is what you expect today about the day after. More generally, E t [E t+1 [y t+j ]] = E t [y t+j ], for j 0. (23) Another implication: the unconditional expectation of conditional expectation is the unconditional expectation itself: E [E t+1 [y t+1 ]] = E [y t+1 ]. (24) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 9 / 45

Why People Care about Short-run Fluctuations Unemployment and in ation, two macro variables that households, rms, and government pay a lot attention, are closely related to short-run uctuations (business cycle uctuations). People are risk averse, so reducing or removing aggregate uctuations may lead to some welfare improvements. Business cycles and nancial systems. Aggregate uctuations and asset prices volatility are also related. Government policies (monetary and scal policies) can a ect aggregate uctuations signi cantly. John Maynard Keynes: In the long run, we re all dead. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 10 / 45

A Special Stochastic Growth Model with Fixed Labor We start with an economy with xed labor supply # subject to max E fc t,k t+1 g " β t u (C t ) t=0, (25) Y t = A t K α t = C t + I t, (26) K t+1 = (1 δ) K t + I t, (27) ln A t+1 = ρ ln A t + (1 ρ) ln A + ε t+1, (28) where the log of the technology shock (A t ) is assumed to be an AR(1) process around the steady state A (normalize A = 1) and ε t+1 is an iid normal variable with mean 0 and variance ω 2. Note that we may assume that ln A t is an AR(1) process around a trend growth path gt: ln A t+1 = ρ ln A t + (1 ρ) gt + ε t+1. Since the trend term does not a ect the model s predictions on business cycle uctuations, we just focus on the simple case (28). Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 11 / 45

y 1 An AR(1) process with ρ=0.95 and σ=0.01 0.8 0.6 0.4 0.2 0 0.2 0.4 0 50 100 150 200 250 300 Period, t Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 12 / 45

Solving the Stochastic Model Just like solving the deterministic model, we can use either dynamic programming (the Bellman equation) or optimal control (the Lagrange multiplier method) to solve this stochastic model. Set up the Lagrangian: " # L = E β t fu(c t ) + λ t [(1 δ) K t + A t Kt α C t K t+1 ]g, t=0 (29) where λ t 0 denote the multiplier on the resource constraint at time t. The FOCs w.r.t. C t, K t+1, and λ t are u 0 (C t ) = λ t, (30) βe t 1 δ + αat+1 Kt+1 α 1 λt+1 = λ t, (31) (1 δ) K t + A t Kt α C t = K t+1 (32) for t = 0, 1,. Note the last FOC is just the resource constraint. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 13 / 45

(Conti.) Combining the above two FOCs yields to the Euler equation: u 0 (C t ) = βe t 1 δ + αat+1 Kt+1 α 1 u 0 (C t+1 ), (33) which together with (32) and the corresponding TVC determine the dynamics of our model economy. To obtain some useful insights about why the optimal growth model can be used to address business cycle uctuations, we rst consider a special case in which u(c t ) = ln C t and δ = 100%. Later we will consider a more realistic model with elastic labor supply and δ < 1 that can be calibrated to the real economy and then used to address aggregate uctuations. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 14 / 45

Special Case with Complete Depreciation When u(c t ) = ln C t and δ = 100%, it is straightforward to show that K t+1 = αβa t K α t, (34) C t = (1 αβ) A t K α t. (35) Taking logs on both sides in the two decision rules: k t+1 = ln (αβ) + a t + αk t, (36) c t = ln (1 αβ) + a t + αk t. (37) where x t = ln (X t ) and a t = ln (A t ) follows the AR(1) process speci ed in (28). Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 15 / 45

Given k 0 and the stochastic process (a t ), (36) and (37) determine the dynamic behavior (i.e., time series) for k t+1 (i.e., i t ) and c t as well as other key macro variables. (e.g., y t = a t + αk t.) Note that this stochastic economy will not converge to a steady state as the disturbance to the technology shock, ε, hits the economy every period and leads to persistent uctuations in consumption, investment, and output. Given the exogenous process (a t ), we can easily compute the stochastic properties of the arti cial model and then compare them with that obtained the real economy (i.e., obtained from the post-war detrended U.S. time series data). Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 16 / 45

Given the simple structure of the model, we can now obtain some key stochastic properties without do simulations. Taking unconditional variances on both sides of (28) (a t+1 = ρa t + ε t+1 ), we have var (a t ) = ω2. 1 ρ 2 Second, taking unconditional variances on both sides of (36) gives ε t var (k t+1 ) = var (1 αl) (1 ρl) 8 " < = : ρ # 9 2= j α k j ; ω2, (38) k=0 k j=0,jk Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 17 / 45

Note that here we use the following fact: k t+1 = ln (αβ) + a t + αk t = ln (αβ) + a t 1 αl ln (αβ) = 1 α + ε t (1 αl) (1 ρl) " k ln (αβ) = 1 α + # ρ j α k j ε t k. (39) k=0 j=0,jk Since y t = a t + αk t, var (y t ) = var (c t ) = var (k t+1 ). (40) Of course, this simpli ed model cannot explain some key stochastic properties observed in the U.S. economy: in the data the log of investment is much more volatile than the log of output, and the log of output is more variable than the log of consumption. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 18 / 45

A Stochastic Growth Model with Elastic Labor Supply We now consider endogenous labor supply in a stochastic growth model and use it to address business cycle uctuations we observed in the real economy. Speci cally, assume that the utility function depends on both consumption (C t ) and leisure (N t = 1 L t ): where L t is labor supply. The following are two popular functional forms: u(c t, N t ), (41) u(c t, N t ) = ln C t + B ln (1 L t ), (42) u(c t, N t ) = ln C t AL t. (43) The latter is based on the indivisible labor assumption and was introduced by Hansen (1985). The speci cation is based on the fact that many workers are employed for a full workweek or they are unemployed. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 19 / 45

Model Setup The optimization problem of the agent is: subject to max E fc t,k t+1 g " β t (ln C t AL t ) t=0 #, (44) Y t = A t K α t L 1 α t = C t + I t, (45) K t+1 = (1 δ) K t + I t, (46) ln A t+1 = ρ ln A t + (1 ρ) ln A + ε t+1, (47) where A, β, δ, α, ρ, and ω 2 are model parameters to be calibrated. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 20 / 45

First-order Conditions Denote Λ t the multiplier on the resource constraint at time t.the FOCs w.r.t. C t, L t, K t+1, and Λ t are where 1 C t = Λ t, (48) A = Λ t (1 α) Y t L t, (49) Λ t = βe t [R t+1 Λ t+1 ], (50) K t+1 = (1 δ) K t + A t K α t L 1 α t C t, (51) R t+1 = 1 δ + α Y t+1 K t+1. (52) Here we write the Lagrange multiplier as Λ t rather than λ t for the consistency with our notational convention to use lower-case letters for log-deviations. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 21 / 45

Steady State A steady state equilibrium for this economy is one in which the technology shock is assumed to be constant so that there is no uncertainty, that is, A t = A = 1, for all t, and the values of capital, labor, and consumption are also constant: K t = K, L t = L, C t = C, Y t = Y. Imposing these steady state conditions in the FOCs, the steady state values can be found by solving the following equations C = 1 α A 1 = β Y L = 1 1 δ + α Y K α A K α L α, (53) = β 1 δ + αk α 1 L 1 α, (54) δk = Y C = K α L 1 α C. (55) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 22 / 45

Calibration Using these steady state equations, we can then choose the model parameters based on long-run growth evidence and econometric studies. This procedure is called calibration. The features of the data which do not exhibit cyclical characteristics are: 1 α=labor s average share of output. β 1 1=average risk-free real interest rate. Given (α, β), choose δ so that the output-capital ratio consistent with the observation. Y is K Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 23 / 45

(Conti.) The parameter A determines the time spent in work activity: Using (53), we have L = 1 α Y A C. Note that (55) implies that Y C = 1/ 1 δ K. Y Hence, using the empirical L, we can calibrate A. Given that the labor share in the postwar U.S. data is 64%, we calibrate α = 0.36. β = 0.99 implies a riskless interest rate of 1%. δ = 0.025 implies the capital-output ratio (where output is measured on a quarterly basis) of roughly 10. A = 3 implies that roughly 30% of time is spent in work activity. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 24 / 45

The Solow Residual We now need to specify the process of the technology shock, i.e., determine the persistence and volatility of the shock, ρ and ω 2. These parameters can be obtained by constructing the Solow residual and then detrending that series linearly. Speci cally, using the production function Y t = A t Kt α Lt 1 α, the Solow residual is de ned as a t = ln A t = ln Y t α ln K t (1 α) ln L t. (56) The a t series can be then be regressed on a linear trend (it is consistent with the assumption of constant technological process) and the residual is identi ed as the technology shock a t. Using the procedure on quarterly data over the U.S. postwar period resulted in ρ = 0.95 and ω 2 = 0.007. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 25 / 45

Log-linearization After calibrating the model parameters, we can now log-linearize the FOCs and the constraint around the steady state. Following the same log-linearization procedure discussed in the last lecture, we introduce lower-case letters to denote log-deviations For example, for the resource constraint K t+1 = (1 δ) K t + Y t C t, we have X t = X exp (x t ). (57) K exp (k t+1 ) = (1 δ) K exp (k t ) + Y exp (y t ) C exp (c t ), K (1 + k t+1 ) = (1 δ) K (1 + k t ) + Y (1 + y t ) C (1 + c t ), Kk t+1 = (1 δ) Kk t + Y y t Cc t, (58) where the constraint is stated in terms of percentage deviations. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 26 / 45

(Conti.) Similarly, 1 C t = Λ t =) c t = λ t, (59) A = Λ t (1 α) Y t L t =) l t = λ t + y t, (60) R t+1 = 1 δ + α Y t+1 K t+1 =) Rr t+1 = α Y K (y t+1 k t+1 ), (61) Λ t = βe t [R t+1 Λ t+1 ] =) λ t = E t [r t+1 + λ t+1 ], (62) Y t = A t K α t L 1 α t =) y t = a t + αk t + (1 α) l t, (63) where a t+1 = ρa t + ε t+1. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 27 / 45

Using the Undetermined Coe cients to Solve the Log-linearized Model Combining with yields y t = 1 α a t + k t + 1 The log-linearized system can be reduced to where α α λ t. 0 = k t+1 + α 1 k t + α 2 λ t + α 3 a t, (64) 0 = E t [ λ t + α 4 k +1 + α 5 λ t+1 + α 6 a t+1 ], (65) α 1 = Y K + (1 δ), α 2 = C K + 1 α α α 4 = 0, α 5 = 1 + (1 α) Y RK, α 6 = Y RK. Y K, α 3 = Y αk, Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 28 / 45

(Conti.) Guess that k t+1 = η kk k t + η ka a t, (66) λ t = η λk k t + η λa a t, (67) where η kk, etc. are just elasticities: If k t = 0.01, i.e., if K t deviates from its steady state by 1%, and a t = 0, then k t+1 = 1% η kk, i.e., K t+1 will deviate from its steady state by η kk percent. Plug the guessed decision rules into (64) and (65) and use the fact that E t [a t+1 ] = ρa t, so that only k t and a t remain: ( η kk + α 1 + α 2 η λk ) k t + ( η ka + α 2 η λa + α 3 ) a t = 0 (68) ( η λk + α 4 η kk + α 5 η λk η kk ) k t + ( η λa + α 4 η ka + α 5 η λk η ka + (α 5 η λa + α 6 )) a t = 0 (69) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 29 / 45

(Conti.) Matching the coe cients. On k t : η 2 kk 0 = η kk + α 1 + α 2 η λk, (70) 0 = η λk + α 4 η kk + α 5 η λk η kk (71) Solving the rst equation for η λk and substitute out in the second equation, we obtain the following quadratic equation: α 2 α 1 η 2 kk which means that η kk = 1 2 η kk + α 1 α 5 = 0, (72) α 4 + 1 α 5 α 5 α 1 + 1α5 η kk + α 1 = 0, (73) α 5 α 1 + 1 α5 r α 1 + 1 α 5 2 4 α 1 α 5 Given that η kk,1 η kk,2 = α 1 α 5 = 1 β = R, at most one root is less than 1 (i.e., at most one stable root). We thus use this root to determine η kk and then determine η λk.!. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 30 / 45

(Conti.) On λ t : 0 = η ka + α 2 η λa + α 3, (74) 0 = η λa + α 4 η ka + α 5 η λk η ka + (α 5 η λa + α 6 ) ρ. (75) Given that we have known η λk, the above equations are just two linear equations in η ka and η λa : η λa = α 3 α 4 + α 3 α 5 η λk + α 6 ρ 1 α 2 α 4 α 2 α 5 η λk α 5 ρ, (76) η ka = α 2 η λa + α 3. (77) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 31 / 45

Numerical Results Using the calibrated parameters and the long-run evidence, Y K = 1 10, C K = 1 13.8, C = 0.72, δ = 0.025, α = 0.36, R = 1.01, Y we can calculate α 1 = Y K + (1 δ) = 1.075, α 2 = C K + 1 α α α 3 = Y αk = 0.278, α 4 = 0, α 5 = 1 + (1 α) Y RK α 6 = Y RK = 0.1. Y K = 0.25, = 1. 063, Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 32 / 45

(Conti.) We now can determine the coe cients in the decision rules: 0 η kk = 1 s 1 @ α 1 + 1α5 α1 + 1 2 4 α 1 A = 0.94,(78) 2 η λk = η kk α 1 = α 2 0.54, (79) η λa = α 3 α 4 + α 3 α 5 η λk + α 6 ρ = 1 α 2 α 4 α 2 α 5 η λk α 5 ρ 0.48, (80) η ka = α 2 η λa + α 3 = 0.158. (81) Using c t = λ t, Using y t = 1 α a t + k t + 1 α α λ t, Using l t = λ t + y t, we have α 5 c t = 0.54k t + 0.48a t. (82) y t = 0.04k t + 1. 924a t. (83) l t = 0.5k t + 1. 44a t. (84) α 5 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 33 / 45

Simulating (or Computing) the Stochastic Properties of the Model Given the exogenous technology process a t+1 = ρa t + ε t+1 and all decision rules obtained above (k t+1, c t, l t, y t ), we can calculate the stochastic properties of the model economy: the volatility, correlation, and persistence (autocorrelation) of all the macro variables (they are also called second moments), and compare them with their empirical counterparts. We can either derive these properties theoretically using the explicit decision rules or use them to simulate the model economy and then compute these properties. Here is the simulation procedure: First, initializing a t = 0, and using a random number generator to generate the innovations to the technology shock, we can create a path for a t. Next, assuming that all remaining values are initially at their steady state (i.e., x t s are zero), the system of the decision rules can be solved to produce time series for all endogenous macro variables. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 34 / 45

(Conti.) We can also derive these stochastic properties theoretically using the resulting decision rules. For example, using we can obtain that k t+1 = η kk k t + η ka a t, (85) a t = ρa t 1 + ε t, (86) k t+1 = η ka a t 1 η kk L = η ka ε t (1 η kk L) (1 ρ L) " # ρ j η k j ε t k, = η ka k=0 k j=0,jk kk (87) Note that the AR(1) process of a t can be written as a t = ε t 1 ρ L = ρ j ε t j. (88) j=0 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 35 / 45

(Conti.) Using (87), we can compute var (k t ), corr (k t, k t+1 ), corr (a t, k t+1 ), corr (y t, k t+1 ), etc, (89) where var () means variance and corr (, ) means correlation (either autocorrelation or correlation with other variables). Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 36 / 45

General Procedures to Compute Second Moments Start with the equilibrium law of motion of the deviation of the state vector: s t+1 = Φs t + ζ t+1, (90) where s t contains rst the endogenous state variables and then the exogenous state variables, and ζ t+1 takes the form:! 0 ζ t+1 =, (91) ε t+1 where ε t+1 denotes the innovation to the exogenous state " vector and!0!0! # 0 distributed with ε t+1 N, Σε. Therefore, Σ ζ =!. 0 Σε Denote Σ s = E [s t s 0 t] the unconditional variance-covariance matrix of s, we have Σ s = ΦΣ s Φ 0 + Σ ζ. (92) The next step is to compute Σ s. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 37 / 45

Method 1 Suppose Φ can be decomposed as Φ = QΛQ 1, where Λ is a diagonal matrix. Let es = Q 1 s, we can write Q 1 s t+1 = ΛQ 1 s t + Q 1 ζ t+1, i.e., es t+1 = Λes t + eζ t+1. (93) De ning Σ es = E [es t es 0 t] and Σ e ζ = Q 1 Σ ζ Q 1 0, we have It follows that Σ es = ΛΣ es Λ 0 + Σ e ζ. (94) Σ es,ij = Σ eζ,ij 1 Λ i Λ j, where i, j = 1, 2,, and Σ es,ij denotes the element of Σ es located in the ith row and the jth column (a similar notation applies for Σ e ζ,ij.) Now we can use Qes = s to obtain: Σ s = QΣ es Q 0. (95) Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 38 / 45

Method 2 We apply for the following iteration method to compute Σ s in the following Lyapunov equation: Σ s,t+1 = ΦΣ s,t Φ 0 + Σ ζ, (96) where Σ s,0 = I. The iteration will stop when Σ s,t does not change much as t increases (i.e., Σ s,t converges to a steady state). Matlab code: lyapcs.m or lyapcsd.m (provided by C.A. Sims) can be used to compute the steady state Σ s. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 39 / 45

Computing Other Second Moments Once the variance-covariance matrix of the state vector has been computed, it is straightforward to nd other second moments of interest. h i Consider E s t st 0 j for j > 0: E s t s 0 t j = E " Φ j s t j + j 1 k=0 = Φ j E s t j s 0 t j = Φ j Σ s. Consider the control vector c t = Πs t : Φ h ζ t h! s 0 t j E c t c 0 t = ΠE st s 0 t Π 0 = ΠΦΠ 0. # More general, E c t c 0 t j = ΠΦ j Σ s Π 0. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 40 / 45

Impulse Response Functions Using s t+1 = Φs t + ζ t+1, (1 ΦL) s t+1 = ζ t+1, where 1 ΦL = 1 + ΦL + Φ 2 L 2 +, we have s t = Φ j ζ t j. j=0 Therefore, IRF (s t+j ) = Φ j ζ t. IRF (c t+j ) = ΠΦ j ζ t. Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 41 / 45

Percent deviation from steady state 2 Impulse responses to a shock in technology output 1.5 labor 1 technology 0.5 consumption capital 0 interest 0.5 0 1 2 3 4 5 6 7 8 Years after shock Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 42 / 45

Percent deviation from steady state 6 Simulated data (HP filtered) output 4 labor 2 0 interest 2 capital consumption 4 6 0 5 10 15 20 25 30 35 40 Year Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 43 / 45

Simulation Results For these variables: capital (k), consumption (c), output (y), labor (l), interest rate (r), investment (i), technology (a), we obtain the following stochastic properties: Autocorrelation, corr (x(t + j), y(t)). (Last row shows j) 0.41 0.30 0.15 0.07 0.35 0.54 0.64 0.68 0.67 0.14 0.02 0.24 0.52 0.87 0.77 0.66 0.54 0.43 0.11 0.27 0.47 0.71 1 0.71 0.47 0.27 0.11 0.20 0.35 0.53 0.74 0.98 0.64 0.37 0.15 0.02 0.23 0.38 0.54 0.74 0.96 0.60 0.32 0.09 0.08 0.17 0.32 0.51 0.73 0.99 0.67 0.40 0.19 0.02 0.12 0.28 0.48 0.72 1.00 0.71 0.46 0.26 0.10 4 3 2 1 0 1 2 3 4 Standard deviations: 0.4950 0.5200 1.8006 1.3721 0.0636 5.7427 0.9260 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 44 / 45

Some Descriptive Statistics for the U.S. Economy and the Model Data Model Consumption c sd (c t ) / sd (y t ) 0.49 0.30 corr (c t, y t ) 0.76 0.87 Investment i sd (i t ) / sd (y t ) 3.02 3.19 corr (i t, y t ) 0.80 0.99 Labor l sd (l t ) / sd (y t ) 0.96 0.76 corr (l t, y t ) 0.88 0.98 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 45 / 45