DSGE Occbin Model Examples FBC Housing AEJ. Macro Topics Universita Cattolica di Milano May 2013 Matteo Iacoviello

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1 Macro Topics Universita Cattolica di Milano May 2013 Matteo Iacoviello

2 We will discuss two broad topics Course overview Solving DSGE models, with particular attention to their nonlinearities Formulating (and estimating) DSGE models with financial frictions, with particular attention to debt and housing

3 DSGE Models: The Simplest Example The planner s problem can be written as: max E t! β s t log C s s=t subject to Optimal consumption implies K t K t 1 = A 1 α t K α t 1 C t (1) 1 1 = βe t C t C t+1 α At+1 K t 1 α + 1!! Assume technology follows an AR(1) process in logs, that is (2) log A t = ρ log A t 1 + log U t (3) where ρ is the autocorrelation of the shock. Assume that log U has mean zero, finite variance.

4 The system made by (1) to (3) is a non-linear system with rational expectations. We usually solve them in the following steps Find the steady state. log A = 0 > A = 1 C = K α 1 1 α 1 αβ 1 α 1 β = αβ > = K K 1 β

5 Linearize around the steady state equation 1 log C t = log 1 C dc t = 1 C C t = A 1 t α Kt α 1 K t + K t 1 A 1 t α Kt α 1 K t + K t 1 take total differential around steady state (1 α) A α K α da t + αa 1 α K α 1 dk t 1 dk t + dk t 1 c t = (1 α) A t + αk t 1 K C k t + K C k t 1

6 equation 2 Ct+1 E t C t = βe t α At+1 K t 1 α + 1! steady state of both sides is α E t c t+1 c t = α (1 α) (E t a t+1 k t ) K (1 α) (1 β) 0 = E t c t+1 + c t + (E t a t+1 k t ) β equation 3 a t = ρa t 1 + u t

7 Taking Stock This is a dynamic system of 3 equations in 3 unknowns. To use a more compact notation, we prefer to write it in the following form where 0 = E t [Fx t+1 + Gx t + Hx t 1 + Lz t+1 + Mz t ] (4) z t = Nz t 1 + e t (5) x t is the vector collecting all the endogenous variables of the model. z t collects all the exogenous stochastic processes. In our above example and F = L = " (1 α)(1 β) β ", G = x = 1 c k 1 # 1 α, M = 0, z = [a] αβ 1 β (1 α)(1 β) β, N = [ρ] # 0 α, H = 1 β 0 0,

8 To summarize, a linearized DSGE model can be written in the following form 0 = E t [Fx t+1 + Gx t + Hx t 1 + Lu t+1 + Mu t ] z t = Nz t 1 + e t The recursive equilibrium law of motion describes endogenous variables as function of the STATE: x t = Px t 1 + Qz t (6) i.e., matrices P, Q such that the equilibrium is described by these rules. Finally, what we do is to plug the matrices in (4) and (5) in a computer, to obtain (6). In our toy example above, set α = 0.33, β = 0.99, ρ = Then ct ct = [z t ] k t k t 1

9 Nonlinear models Nonlinear approximations of structural models can be represented using three equations, written with variables typically expressed in levels. The first equation is s t = f (s t 1, v t ) where s t collects the state variables, v t the collection of structural shocks incorporated in the model. The second equation is the policy function, which incorporates the optimal specification of the control variables c t included in the model as a function of the shocks c t = c (s t ) The third equation maps the full collection of model variables into the observables X t = eg (s t, c t, v t, u t ) where the presence of u t reflects the possibility that the observations of X t are associated with measurement error.

10 Occbin: Toolkit to Solve Models with Occasionally Binding Constraints Occasionally binding constraints arise in many economic applications. Examples include: Models with limitations on the mobility of factors of production; Models with heterogenous agents and constraints on the financial assets available to agents; Models with a zero lower bound on the nominal interest rate; Models with inventory management.

11 Why is a Toolkit Needed? Encompassing realistic features to improve model fit in empirically driven applications may quickly raise the number of state variables. This may render standard global solution methods, such as dynamic programming, infeasible. An alternative that has been used in practice, especially in applications that deal with the zero lower bound on policy rates, is to use a piece-wise perturbation approach. This approach has the distinct advantage of delivering a solution for models with a large number of state variables. Furthermore, it can be easily extended to encompass multiple occasionally binding constraints.

12 Related Stuff Jung, Teranishi, and Watanabe introduced this approach (JMCB, 2005). The approach has been adopted, among others, by Eggertson and Woodford (Carnegie Rochester, 2003), and by Christiano, Eichenbaum, and Rebelo (JPE, 2011). Other methods have been used and proposed. 1) Show a toolbox that extends Dynare to use this solution technique. 2) For simple models, we gauge the performance of the piece-wise perturbation approach relative to the dynamic programming solution.

13 The Solution Method The linearized system of necessary conditions for an equilibrium of a baseline DSGE model can be expressed as: A 1 E t X t+1 + A 0 X t + A 1 X t 1 + Bu t = 0. (M1) There are situations however when one of the conditions describing the equilibrium does not hold, and is replaced by another one.when the "starred" system applies, the linearized system can be expressed as: A 1 E tx t+1 + A 0 X t + A 1 X t 1 + B u t + C = 0 (M2) where C is a vector of constants. Both systems are linearized around the same point same X across systems.

14 The Solution Method When the baseline model applies (M1), we use standard methods to express solution as: X t = PX t 1 + Qu t. (M1_DR) If the starred model applies (M2), shoot back towards the initial condition from the first period when constraint binds again. Main idea: suppose that M1 applied in t 1, M2 applies in t, but M1 is expected to apply in all future periods t + 1, the decision rule in t is: A1 PX t + A0 X t + A 1 X t 1 + C = 0, M2 M1_DR M2 X t = (A1 P + A 0 ) 1 A 1 X t 1 + B u t + C One can proceed in a similar fashion to construct the time-varying decision rules when M2 applies for multiple periods. In each period in which M2 applies, the expectation of how long one expects to stay in M2 affects the value of X t today

15 The Solution Method The search for the appropriate time-varying decision rules implies that for each set of shocks at a point in time one needs to calculate the expected future duration of each "regime." Truncate the simulation at an arbitrary point and reject the truncation if the solution implies that the model has not returned to the reference regime by that point. Start with a guess of the expected durations that is based on the linear solution. Update the guess based on where the conditions of system 1 are violated using the piece-wise linear method until no violation remains.

16 An Example To fix ideas, let s first consider a simple, forward-looking, linear model: where u t is an iid shock. q t = β(1 ρ)e t q t+1 + ρq t 1 σr t + u t r t = max (r, φq t ) The general solution for q t takes the form q t = ε qq,t q t 1 + ε qu,t u t + c q,t r t = ε rr,t q t 1 + ε ru,t u t + c r,t In turn, the ε are functions of q t 1 and u t. How do we find the solution?

17 An Example: Solution ignoring the constraint Ignore the constraint first q t = β (1 ρ) 1 + σφ E tq t+1 + ρ 1 + σφ q t σφ u t q t = ae t q t+1 + bq t 1 + cu t Find solution (method of undetermined coefficients) Match coefficients q t = ε q q t 1 + ε u u t (guess) E t q t+1 = ε q q t (expectation given guess) ae t q t+1 = aε q q t = aε 2 qq t 1 + aε q ε u u t so that eq t 1 + ε u u t = aε 2 qq t 1 + aε q ε u u t + bq t 1 + cu t q t ae t q t+1 ε q = aε 2 q + b, ε u = aε q ε u + c p ε q = 1 1 4ab /2a, ε u = c/ 1 aε q

18 Plug some numbers β = 0.99 φ = 1 ρ = 0.5 σ = 1 r = 0.02 In this case ε q = ε u = so that q t = r t = q t u t

19 Is this solution always correct? Consider the case of a large negative shock to u. If q t 1 = 0, any u t such that r = u < 0.02 u < will violate constraint. Suppose for instance u 1 = 0.2. Ignoring constraint, solution is r t = r t u t r 1 = = r 2 = r 1 = r 3 > 0.02 Hence ignoring the constraint r t would be below 0.02 for 2 periods. Moreover, there is a feedback loop. Higher values of r imply lower q, which implies lower desired values of r, so r can end up being at r for longer

20 We use a guess and verify method to determine how long the constraint will bind. We start by guessing durations that are based on the linear solution that ignores the constraint. Iterate until convergence. So the first guess is going to be 2 periods. > Suppose we guess that r remains at φ for t_low = 2 periods. > Because r is not going to be low as guessed in linear solution... > q will fall more than if r did not hit the constraint... > and r might in turn stay at its lowest bound φ more than t_low periods. In all interesting cases, first guess is not last guess, since dynamics of system depend on feedback loop between duration of constraint and endogenous reaction of variables to constraint. In the example above, one can think of a New Keynesian model at the ZLB.

21 Now cast system using our general notation (use β 0 = β (1 and and β β q t = β 0 E t q t+1 + ρq t 1 σr t + u t r t = max (r, φq t ) ρ)): A 1 E t X t+1 + A 0 X t + A 1 X t 1 + Bu t = 0. (M1) q1 1 σ q ρ 0 ql φ 1 r 0 0 rl 0 r 1 r 1 u = A1 E tx t+1 + A0 X t + A 1 X t 1 + B u t = C (M2) q1 1 σ q ρ 0 ql u = 0 1 r 0 0 rl 0 qt r t X t = PX t 1 + Qu t (M1_DR) εq 0 qt = 1 εu + u ε q 0 r t 1 ε u 0 r 0 0

22 We guess that in period 3 normal system applies. In that case, the solution in period 2 should satisfy where X 2 = (A 1 P + A 0 ) 1 A 1 X 1 + B u 2 + C = P 2 X 1 + Q 2 u 2 + C 2 P 2 = (A 1 P + A 0 ) 1 A 1, Q 2 = (A 1 P + A 0 ) 1 B u 2, C 2 = (A 1 P + A 0 ) 1 C I plug the numbers now. Using A =, P = A =, B 1 = I get X 2 = , C =.023 X , A0 1 1 =

23 We do not know the solution in period 1. However, assuming the starred system applies in t = 1 and is expected to apply in t = 2, the solution in 1 is where A1 (P 2X 1 + C 2 ) + A0 X 1 + A 1 X 0 + Bu 1 + C = 0, X 1 = (A1 P 2 + A0 ) 1 B u 1 + C + A1 C 2 + A 1 X 0 X 1 = P 1 X 0 + Q 1 u 1 + C 1 P 1 = (A1 P 2 + A0 ) 1 A 1 Q 1 = (A1 P 2 + A0 ) 1 B C 1 = (A 1 P 2 + A 0 ) 1 (C + A 1 C 2) After plugging in all the numbers, assuming X 0 = 0, we get X 1 = 0.02

24 Now I plug X 1 back and get X 2 = 0 0 and now I plug X 2 into X 3 which violates constraint in X 3 = PX X 3 = =

25 Note the need to update guess. We guessed that the starred system applies in 2 and that the normal applies in 3. Based on this guess, the starred system applies in 3. Hence we update the guess that starred system applies for 3 periods. Redo the whole thing again until the guessed duration in the starred regime coincides with the actual duration. In the next step, we assume that the normal system applies in 4 but the starred applies in 1, 2 and 3, solve for P 3, Q 3 and C 3, use them to compute X 2 and X 1, and go back to see if X 3 satisfies the constraints. (it does, I have checked it myself)

26 Example 1: Borrowing Constraint Model To check if method works, we apply it to models for which we can compute a a full non-linear solution to arbitrary precision using dynamic programming methods. Consider simple model where a random endowment y t that can be used as collateral u = E 0 t=0 β t log (c t ) c t = y t + b t 1.05b t 1 b t 2y t q log (y t ) = ρ log (y t 1 ) + σ 1 ρ 2 ε t ε t N (0, 1), σ = 0.03, ρ = 0.9 (c1) We look at how solution method handles cases when increases in y t are large enough so that constraint is not binding. We try β = 0.94 and β = Here: constraint (c1) BINDS in normal times.

27 Example 2: RBC with Irreversible Capital Investment cannot fall below a given threshold where φ > 0. u = E 0 t= t log (c t ) c t + k t 0.9k t 1 = A t kt k t 0.9k t 1 φk t 1 (c2) q log (A t ) = 0.9 log (A t 1 ) + σ 1 ρ 2 ε t ε t N (0, 1), σ = 0.03, ρ = 0.9 Here: constraint (c2) DOES NOT bind in normal times.

28 Example 3: Borrowing and Housing U = E 0 t=0 β t (log c t + j log h t ) c t + q t h t = y + b t Rb t 1 + q t h t 1 (1 δ) b t mq t h t log q t = ρ log q t 1 + υ t Here the FOCs would be µ t (b t mq t h t ) = 0 u 0 (c t ) = βre t u 0 (c t+1 ) + µ t Assuming βr < 1, here the borrowing constraint binds in normal times.

29 Structure of Solution Programs (Dynare) The programs we devised take as input two Dynare model files. One.mod file specifies the normal M1 model from which we calculate A 1 E t X t+1 + A 0 X t + A 1 X t 1 = 0. The other.mod file specifies the starred model M2 with the occasionally binding constraint inverted (binding if it was not binding in the reference model, or not binding if it was binding in the reference model). This.mod file yields A 1 E tx t+1 + A 0 X t + A 1 X t 1 + C = 0. We use the analytical derivatives computed by Dynare to construct A 1, A 0, A 1, A 1, A 0, A 1, and C.

30 M1: hp.mod y=1; c+q*h=y+b-r*b(-1)+q*h(-1)*(1-δ); b=m*q*h; lb=1/c-β*r/c(+1); q/c=j/h+β*(1- δ)*q(+1)/c(+1)+lb*m*q; lev=b/(m*q*h)-1; log(q)=ρ*log(q(-1))+u; The main file runsim_houseprice contains M2: hpnotbinding.mod y=1; c+q*h=y+b-r*b(-1)+q*h(-1)*(1-δ); lb=0; lb=1/c-β*r/c(+1)); q/c=j/h+β*(1- δ)*q(+1)/c(+1)+lb*m*q; lev=b/(m*q*h)-1; log(q)=ρ*log(q(-1))+u; 1. mod files: model1 = hp ; model2 = hpnotbinding ; 2. constraint violation triggers switch to m2: constraint= lb<-lbss ; 3. constraint violation triggers switch to m1: constraint_relax1= lev>0 4. solve model1 to obtain M1_DR; write model2 in M2 form 5. given shocks, check if model1 assumptions are violated, if so, look for solution

31 function [zdata zdataconcatenated oobase_ Mbase_] =... solve_one_constraint(model1,model2,... constraint, constraint_relax,... shockssequence,irfshock,nperiods,tol0,maxiter) model1, model2: dynare mod files containing (linear or nonlinear) model equations constraint, constraint_relax: strings with constraints that have to be verified constraint defines the first constraint if constraint is true, solution switches to model2 but if constraint_relax is true, solution reverts to model1 shockssequence: sequence of innovations under which one wants to solve model e.g. randn(100,1)*σ j for simulation or [ 1; zeros(50,1) ] for impulse responses

32 value value value value Simulations - Borrowing and Housing Model House Price Consumption 1.05 Piecewise Linear Nonlinear P erfect Foresight Nonlinear S tochastic Linear Leverage (b/qh) Debt

33 value value value value IRF - Borrowing and Housing Model House Price 1.3 Consumption Piecewise Linear Nonlinear Perfect Foresight Nonlinear Stochastic Linear Leverage (b/qh) Debt

34 Accuracy Borrowing and Housing Model Log Consumption Correlations b qh st.dv skewness ln q, ln c ln q, b qh mean Welf. Linear 6.1% % Occbin 4.5% % Nonlin.perf.fores. 4.6% % Nonlinear stoch. 3.7%

35 A DSGE Model with Multiple Financial Frictions Can financial frictions can explain of the quantitative effects of the financial crisis? Elements of the Financial crisis 1. financial institutions suffer losses which impair their ability to extend credit to the real sector, causing a recession. 2. borrowers balance sheets are impaired, causing a drop in spending 3. credit supply is tight Here we take to the data a model which embeds these elements Model elements: banks and heterogeneous agents Event triggering cycles: (1) financial shocks; (2) changes in asset values; (3) changes in credit supply.

36 Main Findings Financial frictions and shocks in the financial sector account for more than half of the decline in GDP during the last recession Most promising shocks: Declines in asset values ( ) Shocks hitting balance sheet of banks ( ) Tightening of credit standards ( ) We will see how to model them

37 Setup 1. Households. Some households are Savers: buy homes, supply deposits D to banking sector and do not face credit constraints. Some households are Borrowers: borrow L S against their homes. 2. Banks collects deposits from savers and make loans for household borrowers and entrepreneurs. 3. Entrepreneurs borrow from bank, transforms L into K. 4. Competitive firm rents K and N to produce final good Y. HH savers and HH borrowers controlled by wage share in production. Size of Entrepreneurs controlled by capital share in production. 5. Shocks Borrowers are subject to exogenous repayment shocks: they may pay back less agreed Changes in asset values and loan-to-values affect ability to borrow and spend The usual suspects (TFP and preference) Hidden in the background, bells and whistles needed to take model

38 Household Savers Choose consumption and deposits (savings) and hours worked max E 0 t=0 β t H (log C H,t + j t log H H,t + τ log (1 N H,t )) s.t. C H,t + D t + K H,t + q t H H,t = (R M,t + 1 δ K ) K H,t 1 + R H,t 1 D t 1 + W H,t N H,t If these households are not constrained, the equilibrium return on savings will settle around 1/β H in steady state.

39 Household Borrowers Low discount factor, creates simple motive for borrowing β S < β B (s stands for subprime) max E 0 t=0 β t S (log C S,t + j t log H S,t + τ log (1 N S,t )) s.t. C S,t + q t H S,t + R S,t 1 L S,t 1 ε t = L S,t + W S,t N S,t 1 L S,t E t m S,t q t+1 H S,t R S,t If β S is low enough, the constraint on borrowing will hold in a neighborhood of the steady state. ε t is the repayment shock

40 Entrepreneurs Entrepreneurs borrow L E. Transform loan into capital using one-for-one technology, rent capital to representative firm. 1 where β E < γ 1 E βh +(1 γ E ) β 1 B s.t.: max E 0 t=0 β t E log C E,t (β B is banker s discount factor) C E,t + K E,t + R E,t L E,t 1 + q t H E,t = L E,t + (R K,t + 1 δ) K E,t 1 + (R V,t + 1) H E,t 1 L E,t m E,t K E,t + m H E t q t+1 H E,t /R E,t W t N t Borrowing constraint binds if β E R E < 1 which occurs if R E sufficiently low. (R E tied to lenders discount factor)

41 Bankers 1. Bankers transform savings into loans. To do so, they are required to hold some equity (bank capital) in their business 2. Bankers are shortsighted: blinded by greed/impatience, they try and borrow as much as they can from household to increase the size of their balance sheet.

42 The banker s problem Bankers max E 0 t=0 β t B log C B,t where β B < β H, subject to: C B,t + R H,t 1 D t 1 + L E,t + L S,t = R E,t L E,t 1 + R S,t 1 L S,t 1 + D t ε t [ ε t repayment shock ] and additional constraint: D t γ (L E,t + L S,t ε t ) capital adequacy constraint (CAC) CAC forces banker to hold some equity in the business

43 Bank s optimality conditions for D, L E (similar for L S ): 1 λ B,t = E t (m B,t R H,t+1 ) 1 γλ B,t = E t (m B,t R E,t+1 ) Expression for spread: R E,t R H,t = λ B,t m B,t (1 γ E ). λ B : multiplier of bank s capital constraint m B : banker s stochastic discount factor Spread is larger when banker s constraint gets tigher (λ B rises) When constraint gets tighter, bank requires larger compensation on loans to be indifferent b/w making loans and issuing deposits. Loans are more illiquid than deposits: when constraint is binding, a reduction in deposits of 1$ requires cutting back on loans by 1 γ E $. Rise in spread depresses activity when bank net worth is low.

44 Firms Simple Static problem: rent capital from entrepreneurs and hire labor. Production function is: (1 α ν)σ H,t N S,t Y t = K αµ µ) E,t 1Kα(1 H,t 1 Hν α ν)(1 σ) E,t 1N(1 (variable capital utilization allowed for in estimation)

45 Steady State R H = 1 β H return on HH savings β B λ B = 1 β B R H = 1 > 0 banker is constrained β H 1 1 R E R H = (1 γ) > 0 spread β B β H Hence R E > R H (positive banking spreads): 1. Return on bank loans must compensate banker for higher impatience must be higher than cost of deposits to make up for higher liquidity of loans relative to deposits The larger γ, the more loans become substitutes with deposits in the capital adequacy constraint, the lower the extra return on loans required for the bank to be indifferent between borrowing and lending.

46 Calibration 1. Real return on saving 3% per year 2. Capital output ratio about 2.05, Housing output ratio m E, m S, m H : 90%, m K : 50% > Total debt to GDP 105% 4. For added quantitative realism: inertia in the capital adequacy constraint K B,t > ρ B K B,t 1 + (1 ρ B ) ((1 γ) (L E,t + L S,t ε)) quadratic capital adjustment costs

47 Estimation 1. Estimate (with Bayesian methods) µ (capital share of constrained entrepreneurs), ν (share of real estate for entrepreneurs) φ (adjustment costs), AR(1) process for loan losses ε t, parameters describing inertia in borrowing and capital adequacy constraints, curvature utilization function, wage share of constrained HH σ. 2. Use data on Consumption Investment Losses on Loans to Firms Losses on Loans to Households Loans to Households Loans to Firms Housing prices TFP (utilization adjusted measures from Fernald) 3. 8 shocks (housing demand j, financial shocks 1/2 b e, b h, LTV shocks m e, m h, preference p, investment k and TFP shock z) (3 of the shocks are almost perfectly observed)

48 % dev. from trend % of annual GDP % dev. from trend % dev. from trend % dev. from trend % dev. from trend % of annual GDP % dev. from trend Consumption 20 Investment Loan Losses Entr. Loan Losses, HH Loans Entr Loans HH House Prices Technology

49 Remarks 1. I am using net charge-offs for banks from the data, and assuming that these charge-offs apply to the stock of mortgage and non mortgage liabilities of HH and firms (which is larger than the stock of bank loans). Data: cumulative loan losses for comm. banks from 2007 to 2009 around $450bn. Banks own 1/3 of all debt instruments of households/firm: implied losses in the model are 3 times larger. 2. I am assuming that lenders cannot offset future expected charge offs with higher interest rates

50 Financial Shock (blue: baseline model, red: model without banks) Output Consumption Investment Hours House Prices Loans Spread Loan Deposit (pp) Loan Losses RBC Model Frictions E,HH Baseline: Frictions E,HH,BANK

51 YOY % change YOY % change YOY % change YOY % change Historical Decomposition Output Loans Loan Default Shock Housing Demand S hock LTV shock House Prices Investment

52 Historical Decomposition

53 The Timing of the Shocks 1. First stage of financial crisis : Housing Demand Shock drives drop in output 2. Second Stage : Redistribution Shock 3. Third Stage : LTV Shock Estimation tells a story in search of a unifying model (and perhaps one single shock): the decline in housing prices causes defaults which in turn cause tighter credit standard.

54 A DSGE Model of the Housing Market Two questions: 1. What is the nature of the shocks hitting the housing market? 2. How big are spillovers from the housing market to the wider economy? To answer them we build and estimate a quantitative model with: nominal rigidities and monetary policy; multi-sector structure with housing; financing frictions on the household side.

55 Housing and the Macroeconomy

56 Housing and the Macroeconomy

57 Production Model Y sector produces C, IK, intermediate goods (using K and N) IH sector produces new homes (using K, N, land and interm. goods) Different trend progress across sectors (C, IK, IH) Households Patients work, consume, buy homes, rent capital and land to firms and lend to impatient households Impatients/Credit Constrained work, consume, buy homes and borrow against value of home (We set up preferences in a way that borrowing constraint is binding) Sticky prices in the non-housing sector, sticky wages in both sectors Real rigidities: habits in C, imperfect labor mobility, K adjustment costs, variabile K utilization

58 Firms produce : Firms Y t = (A ct N ct ) 1 µ c (z ct k ct 1 ) µ c IH t = (A ht N ht ) 1 µ h µ b µ l (z ht k ht 1 ) µ h k µ b bt lµ l t 1. Y t : non-housing, sticky price sector, IH t flex price sector Two types of households/workers of measure 1 α : wage share of unconstrained households (lenders) 1 α : wage share of constrained households (borrowers) N c = nc α nc 01 α, N h = nh α n01 α h

59 Lenders max E 0 t=0 (βg C ) t z t (logec t + j t log h t τ t g (n ct, n ht )) subject to budget constraint: c t + k ct A kt + k ht + q t (h t (1 δ h ) h t 1 ) + b 0 t = er ct k ct 1 + er ht k ht 1 + R lt l t 1 + Div t + wage 0 t + R t 1b 0 t 1 π t

60 Borrowers Discount future more heavily (β 0 < β) max E 0 t=0 β 0 t G C zt logec t 0 + j t log ht 0 τ t g nct, 0 nht 0 subject to budget constraint c 0 t + q t h 0 t (1 δ h ) h 0 t 1 = wage 0 t + b 0 t R t 1 π t b 0 t 1 and to borrowing constraint b 0 t me t q t+1 h 0 tπ t+1 /R t m : loan-to-value ratio

61 Monetary Policy ry 1 R t = (R t 1 ) r R π r π GDPt t rr G C GDP t 1 u Rt : iid monetary policy shock s t : persistent inflation objective shock rr u Rt s t In Guerrieri and Iacoviello (2013), we allow for ZLB

62 Shocks Stationary AR(1) z t : preference (discount factor) shock j t : housing demand shock (or household technology shock) τ t : labor supply shock u Rt : monetary shock (iid) s t : inflation objective shock u pt : markup/inflation shock (iid) Trend-stationary shocks ln A ct = t ln (1 + γ AC ) + ln Z ct, ln Z ct = ρ AC ln Z ct 1 + u Ct ln A ht = t ln (1 + γ AH ) + ln Z ht, ln Z ht = ρ AH ln Z ht 1 + u Ht ln A kt = t ln (1 + γ AK ) + ln Z kt, ln Z kt = ρ AK ln Z kt 1 + u Kt

63 Model Workings 1. At a basic level, it works like an RBC model with sticky prices/wages in the Y sector, like an RBC with flex prices/sticky wages in the IH sector (added twist: IH sector produces durables) 2. Sector specific shocks or preference shocks can shift resources from one sector to the other 3. Housing collateral generates wealth effects on consumption

64 Trends 1. Log preferences and Cobb-Douglas yield balanced growth 2. C and qih grow at the same rate over time. 3. IK can grow faster than C, thanks to A K progress 4. IH can grow slower than C, if land is a limiting factor and A H is slow 5. Long-run growth rates C C IK IK = γ AC + µ c 1 µ c γ AK = γ AC µ c γ AK IH IH = (µ h + µ b ) γ AC + µ c (µ h + µ b ) 1 µ c γ AK + (1 µ h µ l µ b ) γ AH q q = (1 µ h µ b ) γ AC + µ c (1 µ h µ b ) 1 µ c γ AK (1 µ h µ l µ b ) γ AH

65 2. Estimation 1. Use 10 time-series (1965Q1-2006Q4) for US per capita C, IH, IK, real housing price q R, π, sectoral hours N c and N h, sectoral wages w c and w h 2. Some parameters calibrated to match steady state ratios β = , β 0 = 0.97, m = 0.85 Y = Nc 0.65 kc 0.35, IH = Nh 0.70 kh 0.10 kb 0.10 l 0.10 Targets: (K + qh) /GDP = 3.2, (qh) /GDP = 1.35, (δ h qh) /GDP = Other parameters (including degree of financing frictions) estimated by Bayesian techniques

66 3. Results Prior and Posterior Parameters 1. Slow rate of technological progress in housing construction (γ AC = 0.32%, γ AH = 0.08%) 2. Wage share of credit constrained households 1 α =21 percent 3. High price rigidity (θ π = 0.83) and indexation (ι π = 0.71) High wage rigidity (θ wc = 0.81, θ wh = 0.91), low wage indexation (ι wc = 0.07, ι wh = 0.42) 4. Taylor rule: R t = 0.61R t [1.38π t (gdp t gdp t 1 )]

67 Variance Decomposition Housing demand shocks and housing technology shocks account for one quarter each of the volatility of residential investment and house prices. Monetary shocks account for about 20 percent

68 Impulse Responses, Housing Preference Shocks 0.15 Real Consumption 0.1 Real Business Investment Real Residential Investment 1 Real House Prices Flexible Wage 0.2 No Collateral Effects 0Baseline

69 Impulse Responses, Monetary Shocks 0 Real Consumption 0.5 Real Business Investment Real Residential Investment 0.5 Real House Prices Flexible Wage 0.5 No Collateral Effects Baseline

70 Role of Monetary Shocks 1. Sensitivity of residential investment to monetary shocks larger than that of business investment, in line with VAR evidence 2. Key reason: wage stickiness If IH sector were flex wage, flex price, it would not contract after contractionary policy (BHK 2007) 3. Model elasticity of house prices to a monetary shocks of similar magnitude to what is found in VAR studies

71 Our two original questions, revisited. 1. What drives the housing market? Focus on recent period. 2. How big are the spillovers? Focus on pre and post 1980 s

72 Focus on : Drivers of Housing Market Period % q Technology Monetary Pol. 1998:I 2005:I :II 2006:IV % IH 1998:I 2005:I :II 2006:IV Comparison with period: monetary policy has played a larger role here.

73 Size of Spillovers Most of the spillovers are through the effect on consumption. For given LTV m, they are a function of α. Regression based on artificial data generated by the model log C t = log HW t 1 if α = 0.79 log C t = log HW t 1 if α = 1 To better measure spillovers in sample, we re-estimate the model across subsamples ( , ). First period: fix m = 0.775, 1 bα = 0.33 Second period: fix m = 0.925, 1 bα = 0.21 Two implications Monetary policy more powerful in the second period Housing shocks have larger spillover effects on consumption in the second period