Vector Autoregressions as a Guide to Constructing Dynamic General Equilibrium Models

Transcription

1 Vector Autoregressions as a Guide to Constructing Dynamic General Equilibrium Models by Lawrence J. Christiano, Martin Eichenbaum and Robert Vigfusson 1

2 Background We Use VAR Impulse Response Functions as Guide to Constructing Monetary DSGE Models Basic Strategy: Choose Model Parameters to Make Model-based Impulse Responses as Close as Possible to VAR-based Impulse Responses. Problem: The Usefulness of Vector Autoregressions Has Been Questioned For Example: CKM Question the Use of VARs with Long-Run Restrictions... CKM Argue (as do Others) That First Differencing Hours Worked When Hours are Stationary Can Lead to Incorrect VAR Inference (DSVARS) Here, We are Concerned With their Findings for LSVARS, When Hours are Correctly Specified in Levels CKM Report Two Major Findings. VARs Increasingly Distorted and Misleading as Datasets Grow We Do Not Find This. VARs Are Uninformative in Small Samples. We Disagree. 12

3 Findings All Findings Based on Running VARs in Data Generated by DSGE Models. Long-Run Exclusion and Sign Restrictions Our Examples: As Long As there are Enough Variables in the Analysis - Inference from VARs Reliable. There May be Substantial Sampling Uncertainty But, A Conceptual Error Leads CKM to Greatly Overstate that Uncertainty. In Any Case, VAR Methods Accurately Characterize Uncertainty. A User of VARs Would Not Be Misled in Practice Short-Run Exclusion and Sign Restrictions Our Examples Suggest No Evidence of Small-Sample Bias Relatively Little Sampling Uncertainty Robust to Which Variables are Included in VAR 2

4 Outline Analysis Based on Simple RBC Model Long Run Restrictions Short Run Restrictions Analysis Based on Monetized RBC Models With Various Frictions (ACEL) 18

5 DGP: A Generic RBC Model Preferences: E X t= (β (1 + γ)) t log c t + ψ log l lt. Constraints: c t +(1+τ x,t )[(1+γ) k t+1 (1 δ) k t ] (1 τ lt ) w t l t + r t k t. Shocks: c t +(1+γ) k t+1 (1 δ) k t k α t (Z t l t ) 1 α. log (Z t )=µ Z +log(z t 1 )+σ z ε z t, τ lt+1 =(1 ρ l ) τ l + ρτ lt + σ l ε d t+1, τ xt+1 = ρ x τ xt + σ x ε x t+1. 22

6 Quarterly Model: CKM Parameter Values Nonstochastic Part /4 5/4 Parameters Estimated by Maximum Likelihood: log ( )=154+log( 1 ) =(1 938) Following Parameters Vary Across Examples. +1 =

7 Alternative Parameterizations of RBC Model Prescott Version (One Important Shock) log ( )=154+log( 1 )+1 +1 = = CKM Benchmark (Two Important Shocks) log ( )=154+log( 1 ) =(1 938) = 22

8 Alternative Parameterizations of RBC Model... Three Shocks, Two Important log ( )=154+log( 1 ) =(1 938) =1 Three Important Shocks log ( )=154+log( 1 ) =(1 938) =

9 Using VARs to Estimate the Effects of a Positive Technology Shock VAR: = = =, = = = log log =[ ] = 1 2 = 3 Impulse Response to Positive Technology Shock ( 1 ): 1 = = = Need:

10 Identification Problem From Applying OLS To Both Equations in VAR, We Know : 1 Problem, Need first Column of 1 Restrictions (Bivariate Case): three equations in four unknowns = Identification Problem: Not Enough Restrictions to Pin Down 1 Need More Restrictions 25

11 Identification Problem... Two Key Properties of DGP: Long-Run Exclusion Restriction: lim [ ]=( only) Sign Restriction: increasing in These Properties Provide Sufficient Restrictions to Pin Down 1 26

12 Characterizing Restrictions Note: [ +1 ] 1 [ +1 ]=[ ]+[ 1 ] Then ( =1) [ +1 ] 1 [ +1 ]=(1 ) [ + ] [ +2 ] 1 [ +2 ]=(1 ) [ + ] 1 [ + ]=(1 ) as : lim [ + ] 1 [ + ] = lim (1 ) =(1) [ ] 1 29

13 Characterizing Restrictions... As j (for arbitrary p) : lim j E t[a t+j ] E t 1 [a t+j ]=(1, ) [I B(1)] 1 Ce t 37

14 Characterizing Restrictions... The VAR: X t = B 1 X t 1 + B 2 X t B p X t p + u t Identification: Solve for C Such that - (exclusion restriction) (1, 2) element of [I B(1)] 1 C = number,..., numbers numbers (sign restriction) (1, 1) element of [I B(1)] 1 C is positive CC = V There Are Many C That Satisfy These Constraints. All Have the Same C 1. 38

15 Let Proof of Uniqueness of C 1 D [I B(1)] 1 C so, DD = [I B(1)] 1 V [I B(1) ] 1 S (Since CC = V ) Exclusion Restriction Requires: D = d11 D 21 D 22 So d DD 2 = 11 d 11 D21 D 21 d 11 D 21 D21 + D 22 D22 = Sign Restriction: d 11 >. Then, First Column of D Uniquely q Pinned Down: d 11 = S 11,D 21 = S 21 /d 11 First Column of C Uniquely Pinned Down: C 1 =[I B(1)] D 1. S 11 S 21 S 21 S

16 Proof of Uniqueness of C 1... Remark: Sign of C 11 (Impact Effect of Technology Shock) unrestricted (1, 1) Element of [I B(1)] 1 C Must Be Positive (Long Run Effect) Could Lead to Contemporaneous Drop in Productivity 41

17 Proof of Uniqueness of C 1... Remark: Sign of C 11 (Impact Effect of Technology Shock) unrestricted (1, 1) Element of [I B(1)] 1 C Must Be Positive (Long Run Effect) Could Lead to Contemporaneous Drop in Productivity Remark: CKM Sign Restriction: C 11 >, (1, 1) Element of [I B(1)] 1 C unrestricted Positive Technology Shock Leads to Contemporaneous Rise in Productivity Positive Technology Shock Could Lead to Permanent Reduction in Productivity This Pattern is Impossible in CKM DGP 42

18 Assessing the VAR With Long-Run Restrictions Using RBC as DGP Experiments Simulate 5 Artificial Data Sets, Each of Length 18 From Various Versions of DGP Estimate VAR On Each Artificial Data Set and Compute Dynamic Response of Hours to Positive Technology Shock Report Mean Impulse Response Function and Plus/Minus Two Standard Error Bounds (Grey Area) 39

19 Assessing the VAR With Long-Run Restrictions Using RBC as DGP... General Findings: If there are Enough Variables, Relative to the Number of Important Shocks to the Economy, then VAR Reliable Precision May be Low, But VAR Will Tell You When this is So. Often, But not Always, One Can Discriminate Between Interesting Hypotheses CKM Finding of Enormous Sampling Uncertainty Reflects CKM s Mistaken Sign Restriction We are Less Pessimistic than CKM About What Happens as More Data Come In (These Findings are Somewhat Tentative, Until We Fully Understand Why Our Results Differ from CKM.) 42

20 Notes on Figures and 1 Figure : Prescott RBC Model with Standard Deviation,.1, on Technology Shock Other Shocks Tiny: +1 = +1 = Range of Impulse Responses Small If the Model is True, Econometrician Would Reject Interesting Alternatives (e.g., King-Wolman and Francis-Ramey Models in Which Hours Fall After Technology Shock). This presumes that +/- 2 standard deviation bounds in Figure corresponds to confidence intervals that an econometrician would find if the model were true. In fact, Figure 2 indicates that this is a reasonable approximation. Fig 2 reports the actual standard deviation of econometrician s estimator of the impulse response function (light blue line with dots) as well as the mean (and plus/minus two standard error bands) of the econometrician s bootstrap estimator of standard 43

21 Notes on Figures and 1... deviation. Impulse Response Functions Estimated Very Tightly Can Easily Exclude Interesting Alternatives 44

22 Notes on Figures and 1... Figure 1: 1,1 Element: Two Variables, One Important Shock CKM Estimated Technology Shock Process Other Shocks Tiny: +1 = +1 = Same as in Figure 1,2 Element: Two Variables, Two Important Shocks - Benchmark CKM Example CKM Estimated Technology and Labor Tax Shocks +1 = Substantial Small Sample Bias Sampling Uncertainty Reported Here VERY Different from That Reported for Same Example in CKM (Will Return to this Below). 45

23 Notes on Figures and ,1 Element: Three Variables, Two Important Shocks CKM Estimated Technology and Labor Tax Shock Process Investment Price Shock: +1 =1 +1 VAR Bias Disappears. Sampling Uncertainty Higher, Harder to Reject Interesting Hypotheses 2,2 Element: Three Variables, Three Important Shocks CKM Estimated Technology and Labor Tax Process CKM Price of Investment Example: +1 = Small Sample Bias Reappears 46

24 Notes on Figures and ,2 Element: Four Variables, Three Important Shocks Add to Data in VAR, Plus Additive Measure Error with.1 Standard Deviation CKM Estimated Technology and Labor Tax Process CKM Price of Investment Example. Bias is Gone, But Sampling Uncertainty is Substantial. 47

25 Figure : Bivariate VAR, Prescott RBC Model Mean, Small Sample VAR True Plus, Minus 2 Standard Deviations Quarters Percent Response in Hours to One-Standard Deviation (1 Percent) Technology Shock

26 Bivariate VAR Small: Two Variables, One Important Shock Variables Small: Three Variables, Two Important Shocks Variables CKM: Four Variables, Three Important Shocks Bivariate VAR CKM: Two Variables, Two Important Shocks Variables CKM: Three Variables, Three Important Shocks

27 1.5 Bivariate VAR Small 1.5 Bivariate VAR CKM Variables Small Variables CKM Variables CKM

28 What We Learn From the Examples in Figures and 1: Tentative Conclusion: VAR Inference With Long-Run Identification Not Misleading As Long as there Are Enough Variables in the Analysis. Need at Least One More Variable than the Number of Important Shocks Driving the Data In Practice: Roughly 4-5 Variables (see Sargent-Sims, Quah-Sargent, Uhlig ( What moves real GDP? ), who show that 3-4 shocks account for most fluctuations; see Prescott (1986) who argues 1 shock account for 7% of fluctuations.). 5

29 How Big is the Sampling Uncertainty Associated with Long-Run Identifying Restrictions? CKM Provide a Misleading Answer to this Question Problem: CKM Sign Restriction Leads Them To Confuse Positive and Negative Technology Shocks 51

30 How Big is the Sampling Uncertainty Associated with Long-Run Identifying Restrictions?... Benchmark CKM Model Initial Hours Worked Response Strong, Productivity Response Weak (Fig 3) 52

31 Response of Hours Worked and Labor Productivity in Benchmark CKM Example.65 Labor Productivity Hours Worked

32 How Big is the Sampling Uncertainty Associated with Long-Run Identifying Restrictions?... Benchmark CKM Model Initial Hours Worked Response Strong, Productivity Response Weak (Fig 3) Consequences of CKM Sign Restriction In Some Samples, Hours Response Very Strong and Initial Productivity Small Negative CKM Algorithm Calls this a Negative Technology Shock In These Cases, Their Procedure In Effect Multiplies the Hours Response by -1 (Fig 3a). 53

33 Figure 3a: Example of Mistaken Inference.6 productivity (LR).2 productivity hours (LR) hours

34 How Big is the Sampling Uncertainty Associated with Long-Run Identifying Restrictions?... Benchmark CKM Model Initial Hours Worked Response Strong, Productivity Response Weak (Fig 3) Consequences of CKM Sign Restriction In Some Samples, Hours Response Very Strong and Initial Productivity Small Negative CKM Algorithm Calls this a Negative Technology Shock In These Cases, Their Procedure In Effect Multiplies the Hours Response by -1 (Fig 3a). Is this a Theoretical Curiosum? No...It Happens 23% Of Time. Leads to Pronounced Bimodal Distribution of Impact Effects (upper panel, Fig 4) 54

35 1 Hours Impact.1 Labor Productivity Response

36 How Big is the Sampling Uncertainty Associated with Long-Run Identifying Restrictions?... Benchmark CKM Model Initial Hours Worked Response Strong, Productivity Response Weak (Fig 3) Consequences of CKM Sign Restriction In Some Samples, Hours Response Very Strong and Initial Productivity Small Negative CKM Algorithm Calls this a Negative Technology Shock In These Cases, Their Procedure In Effect Multiplies the Hours Response by -1 (Fig 3a). Is this a Theoretical Curiosum? No...It Happens 23% Of Time. Leads to Pronounced Bimodal Distribution of Impact Effects (upper panel, Fig 4) Actual Distribution of Impact Effects Not Bimodal (lower panel, Fig 4). 55

37 1 Hours Impact.1 Labor Productivity Response Hours Impact 1 x 1 3 Labor Productivity Response

38 How Big is the Sampling Uncertainty Associated with Long-Run Identifying Restrictions?... Benchmark CKM Model Initial Hours Worked Response Strong, Productivity Response Weak (Fig 3) Consequences of CKM Sign Restriction In Some Samples, Hours Response Very Strong and Initial Productivity Small Negative CKM Algorithm Calls this a Negative Technology Shock In These Cases, Their Procedure In Effect Multiplies the Hours Response by -1 (Fig 3a). Is this a Theoretical Curiosum? No...It Happens 23% Of Time. Leads to Pronounced Bimodal Distribution of Impact Effects (upper panel, Fig 4) Actual Distribution of Impact Effects Not Bimodal (lower panel, Fig 4). How Does this Affect Standard Errors? Leads CKM to Substantially Overstate Sampling Uncertainty (Fig. 5) 56

39 Impact of CKM Sign Restriction on Sampling Uncertainty 1.5 Long-run Identification 1.5 CKM Identification

40 Would the Econometrician Be Misled in Large Samples? CKM Say their Benchmark Example Implies Yes We Say No. In Practice, Econometrician Applies Diagnostic Tests for Lag Lengths in VAR Four Monte Carlo Simulations of Artificial Data. Simulate 5 Datasets of length =1; 5 ; 2 ; 5 In Each Artificial Data set We Choose Lag Length, as Solution to min () 445 where () is Akaike criterion. Compute Mean Impulse Response Across 5 Data sets (Fig 6) 6

41 .7.6 T=1 N =5.2 T=5 N=9.8 T=2 N=14.8 T=5 N=18.5 Model

42 Would the Econometrician Be Misled in Large Samples? CKM Say their Benchmark Example Implies Yes We Say No. In Practice, Econometrician Applies Diagnostic Tests for Lag Lengths in VAR Four Monte Carlo Simulations of Artificial Data. Simulate 5 Datasets of length =1; 5 ; 2 ; 5 In Each Artificial Data set We Choose Lag Length, as Solution to min () 445 where () is Akaike criterion. Compute Mean Impulse Response Across 5 Data sets (Fig 6) Key Result: Mean is Converging to Right Solution. Unresolved Puzzle: We Get Faster Convergence As Lag Length Increases (see analytic Plims, Fig 7) 61

43 .7 Figure 7: Response of Hours Worked, Analytic Plims, for Lag Length p = 4, 8, 16, 32; True Response *

44 Assessing the VAR With Short-Run Restrictions Using RBC as DGP Experiments Simulate 5 Artificial Data Sets, Each of Length 18 From Various Versions of DGP Estimate VAR On Each Artificial Data Set and Compute Dynamic Response of Hours to Positive Technology Shock Report Mean Impulse Response Function and Plus/Minus Two Standard Error Bounds (Grey Area) Timing in Two Shock Model: Labor Tax Shock Realized, Labor Determined, Technology Shock Realized, Consumption, Investment Determined. 62

45 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Note: log = 1 log = ( ) so least squares regression has error µ log ; log log = Equivalent to VAR With Choleski Ordering: = = =, = = lower triangular µ µ log 1 = log = 2 2 technology shock Everything is Perfect: No Bias and Small Standard Errors (Fig 8) 65

46 Two Variables in VAR Benchmark CKM Model With Timing Restriction Percent Response Three Variables in VAR, Investment Shock Not Persistent (Two Important Shocks) Percent Response Three Variables in VAR, Investment Shock Persistent (Three Important Shocks) Percent Response Periods After the Shock

47 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Timing in Three Shock Model: Labor Tax Shock Realized, Labor Determined, Technology and Investment Price Shocks Realized, Consumption, Investment Determined. 67

48 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Note: log = 1 1 log = ( 1 ) log = ( 1 ) so least squares regression has error log = µ log ; log log + 68

49 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Equivalent to VAR With Choleski Ordering: = = =, = = lower triangular = log log log = technology shock Everything is Perfect: No Bias and Small Standard Errors, Easy to Rule Out Interesting Hypotheses (Fig 8) Standard Error Estimates Unbiased (Fig 9) 69

50 Percent Response Percent Response Percent Response Two Variables in VAR CKM Benchmark Three Variables in VAR, Investment Shock Not Persistent (Two Important Shocks) Three Variables in VAR, Investment Shock Persistent (Three Important Shocks) Periods After the Shock

51 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Timing in Three Shock Model: Labor Tax Shock Realized, Labor Determined, Technology and Investment Price Shocks Realized, Consumption, Investment Determined. 67

52 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Note: log = 1 1 log = ( 1 ) log = ( 1 ) so least squares regression has error log = µ log ; log log + 68

53 Assessing the VAR With Short-Run Restrictions Using RBC as DGP... Equivalent to VAR With Choleski Ordering: = = =, = = lower triangular = log log log = technology shock Everything is Perfect: No Bias and Small Standard Errors, Easy to Rule Out Interesting Hypotheses (Fig 8) Standard Error Estimates Unbiased (Fig 9) 69

54 Percent Response Percent Response Percent Response Two Variables in VAR CKM Benchmark Three Variables in VAR, Investment Shock Not Persistent (Two Important Shocks) Three Variables in VAR, Investment Shock Persistent (Three Important Shocks) Periods After the Shock

55 Key Lessons of the RBC Model Analysis With Short Run Exclusion Restrictions, VAR Analysis Highly Accurate With Long Run Exclusion Restrictions: Potential Problems with Large Sampling Uncertainty Property of Our Examples: Conditional on Having Enough Variables, the Analyst Would Know it. 69

56 Motivation for Additional Analysis We Use VAR Impulse Response Functions as part of a Strategy To Formulate and Estimate Monetary DSGE Models Basic Strategy: Choose Model Parameters to Make Model-based Impulse Responses as close as Possible to VAR-based Impulse Responses. Problem: How Reliable are VAR-Based Impulse Response Functions? Solution: Extend Strategy Pursued Above to Larger DSGE Models, Larger VARs and More Shocks and Variables. Run the VAR in Data Generated by the Model. We Look at this in ACEL. 7

57 ACEL Model Final Good Firms: Z 1 () () () Z 1 = () 1 71

58 ACEL Model... Intermediate Good Firm: max X = + { + () + () () ()+ ( + ()) () + } technology: () given () = () ( ()) 1 µ +1 () =(1) () ()+(1 ) () 1 () = = in steady state 72

59 ACEL Model... Calvo Pricing: () : ½ set optimally with probability 1 = 1 1 () with probability Labor Market: Perfectly Competitive Labor Contractor Hires Differentiated Labor Services, and Sells Homogeneous Labor: max Z 1 subject to: = Z

60 ACEL Model... Household... Preferences: " # X 2 + log ( + +1 ) 2 = Asset Evolution Equation: +1 = [ +( 1) ]+ + + (1 + ( )) = : Beginning of Period Base Money; : Transactions Balances 74

61 ACEL Model... Monopoly Supplier of Specialized Labor Input, : ½ set optimally = 1 1 with probability 1 with probability Market Clearing: Loan Market: = +( 1) Resource Constraint: Z 1 (1 + ( )) + 1 ()+( ()) () 75

62 ACEL Model... Exogenous Shocks: Disembodied Technology: 1 = ˆ = ˆ 1 + ˆ Investment specific: 1 = ˆ = ˆ 1 + Balanced Growth: 1 = 1 = 76

63 ACEL Model... Monetary Policy: +1 =1+ ˆ = ˆ +ˆ +ˆ ˆ = ˆ 1 + ˆ = ˆ ˆ = ˆ

64 ACEL Model... Equilibrium Inflation: ˆ = ˆ +1 + ˆ Parameter, : 1 1 = ( ) ½ 31 Firm-specific Capital ( =2 =11 =36) = 1 Homogeneous Capital 79

65 MC 1,f P 1 MC,f P 2 P B MC 1 B A MC Q Q

66 ACEL Model... Intuition About Importance of Firm Specificity of Capital A Firm Contemplates Raising Price This Implies Output Falls Marginal Cost Falls Incentive to Raise Price Falls Effect Quantitatively Important When: Demand Elastic Marginal Cost Steep 8

67 Estimating Parameters in the Model Partition Parameters into Three Groups. Parameters set a priori (e.g.,...) 1 : remaining parameters pertaining to the nonstochastic part of model 1 =[ ] 2 : parameters pertaining to stochastic part of the model Number of parameters, =( 1 2 ) to be estimated - 18 Estimation Criterion () :mapping from to model impulse responses ˆ : 592 impulse responses estimated using VAR Estimation Strategy: ³ ˆ ³ = arg min ˆ () 1 ˆ () : diagonal matrix with sample variances of ˆ along the diagonal. 81

68 Estimated VAR {z} 1 1 = ln (relative price of investment ) ln ( Hours ) ln ( deflator ) Capacity Utilization ln (Hours ) ln ( Hours ) ln ( ) ln ( ) ln ( ) Federal Funds Rate ln( deflator )+ln( ) ln ( ) 82

69 Estimated Parameter Values, a little low... similar to other estimates in literature wages reoptimized on average every 3.6 quarters Parameters important in subsequent discussion... costly to vary utilization of capital close to perfect competition amazingly low! (similar to estimates reported in literature) 83

70 Estimated Parameters of Exogenous Processes, 2 Shocks ˆ = 24 (22)ˆ =3 (4) ˆ = 9 (11)ˆ =7 (4) Monetary Policy ˆ = ˆ +ˆ +ˆ ˆ = 3ˆ =33 (11) (8) ˆ = 33 (25)ˆ 1 +3 (192) +142 (99) 1 ˆ = 82 (12)ˆ (21) +13 (21) 1 84

71 Results Monetary Policy Shocks Substantial Propagation in Response to Policy (R and M move for only a year) Enormous Inertia in Inflation No Noticeable Rise in Inflation for a Year In Model, Prices Reoptimized Frequently. Persistence in Response of Output Neutral Technology Shocks Model Does Well, Although Confidence Intervals Wide Model Failure: Inflation Responds Strongly in Data, Not Strongly in Model Positive Response of Hours to Technology Shock is Due to Monetary Accommodation. Embodied Technology Shocks The model does well on these. 85

72 Figure 1: Response to a monetary policy shock (o - Model, - VAR, grey area - 95 % Confidence Interval) Output MZM Growth (Q) Inflation Federal Funds Rate Capacity Utilization Average Hours Real Wage Consumption Investment Velocity Quarters.2.1 Investment Good Price Quarters 2 Total money growth (M) Quarters

73 Figure 2: Response to a neutral technology shock (o - Model, - VAR, grey area - 95 % Confidence Interval) Output MZM Growth (Q) Inflation Federal Funds Rate Capacity Utilization Average Hours Real Wage Consumption Investment Velocity Quarters Investment Good Price Quarters 1.5 Total money growth (M) Quarters

74 Figure 3: Response to an embodied technology shock (o - Model, - VAR, grey area - 95 % Confidence Interval) Output MZM Growth (Q) Inflation Federal Funds Rate Capacity Utilization Average Hours Real Wage Consumption Investment Velocity Quarters Investment Good Price Quarters.5 Total money growth (M) Quarters

75 Question We fit Impulse Responses Reasonably Well, But Do We Have Any Reason to Trust Them? Obvious Experiment: What if the Estimated Model were literally true, How Well Would Our Procedure Work? We Plan to Study the Sampling Distribution of our Parameter Estimator For Now, We Repeated the RBC analysis Described Above. 86

76 Plan Basic Idea: Generate Data from DSGE Model and Fit VARs in Artificial Data Problem: DSGE Model Has Too Few Shocks Empirical Procedure Recognizes We re Short on Shocks and Offers a Natural Solution 87

77 Background The VAR: = () 1 + = () Stochastic Process for Can Be Decomposed Into Two Orthogonal Pieces: = + = () = ()

78 Background... = + Piece, Is Captured By the Equilibrium Model Piece, Is Left Out. Implications of the Analysis: Data Generated From Model Corresponds to Data Generated From Model is Missing. To Run VAR in Artificial Data Generate Artificial Data From Economic Model, Generate Construct: Fit VAR to From = = () µ

79 Experiment Generate Artificial Data Extremely Long Data Set to Get Plim (2, Observations) Many Data Sets of Length 17 Each Feed Each Data Set to Same VAR Fit to US Data Compute Impulse Response Functions Dotted Lines: Small Sample Means Dashed Lines: Plims 91

80 Findings Contents of Figures x s - mean impulse response, plus/minus two times mean standard deviation estimated by econometrician Solid line: truth Dotted line: small sample mean from VAR Gray area: mean impulse response, plus/minus two times actual standard deviation of impulse response. 92

81 Findings... Monetary Policy Shock Bias Reasonably Small Estimator of Uncertainty Acceptable (Though Some Bias Down) Neutral Technology Shock Overall, Bias Small, But Output, Investment and Real Wage Biased Down Notable Finding: Inflation Response Rules Out Estimated VAR Inflation Response VAR Confidence Interval: -.22 to -.8 percent Evidence Against ACEL Model. Evidence of Potential Power of Long-Run Restrictions. Embodied Technology Shock Performs Reasonably, Overall 93

82 Output Fed Funds Real Wage Velocity M2 Growth Capacity Util Consumption Price of Inv Inflation Avg Hours Investment

83 .4.2 Output M2 Growth.2.2 Inflation Fed Funds Real Wage Capacity Util Consumption Avg Hours Investment Velocity Price of Inv

84 .4.2 Output M2 Growth Inflation Fed Funds.5 Capacity Util.4 Avg Hours Real Wage Consumption Investment Velocity Price of Inv

85 Conclusion VARs With Long-Run Restrictions: Can Be Imprecise, But Standard Errors Let You Know Can Have Power See Prescott RBC Model See Evidence Against ACEL Model VARs With Short-Run Restrictions: Appear to Be Precise Simulations Like the Ones Done Here (following Erceg-Guerrieri-Gust, and CKM) Important as Ultimate Guard Against Being Misled However, in Models Which Do Not Specify All Shocks, Must Use Something Like ACEL Procedure to Add Estimate of Other Shocks. 96