1. Ten Years after the Financial Crisis:   What Have We Learned from the Renaissance in Fiscal Research?by
Valerie A. Ramey
University of California, San Diego and NBER
Melbourne Institute Macroeconomic Policy Meetings, October

2. Progress on All 3 Methodological Fronts
Theory
Incorporation of sticky prices & wages, hand-to-mouth consumers, lower bounds on policy interest rates, currency unions, variety of financing methods, effects of anticipated fiscal policy, debt sustainability.
2. Empirical methods
Identification via natural experiments, narrative methods, Bartik-style instruments; proxy SVAR/external instrument methods, local projection methods for estimating dynamic responses, standardization of methods for computing multipliers, incorporation of state dependence.
3. Data
Newly constructed historical and cross-sectional data sets within countries, narrative instruments for panels of countries, exploitation of the rich new data created by the variety of policymakers’ fiscal responses to the crisis.

3. Outline Summary of leading empirical approaches.
Strengths and weaknesses.
Some innovations and lessons learned.
2. Summary of estimates at this point in time.
3. What do household and cross-sectional estimates tell us about national multipliers?
4. Conclusions.
The higher the mpc, the higher the multiplier.

4. 1. Summary of Leading Empirical Approaches

5. A. Strengths and Weaknesses
Type
Advantages
Disadvantages
Identification of exogenous policy shocks is often challenging.
Estimates are based on historical happenstance.
Difficult to net out other fiscal changes.
It is difficult to construct counterfactuals.
1. Aggregate country-level
time series evidence
Estimates are directly informative about national-level multipliers.
2. DSGE models, estimated or calibrated
Estimates are directly informative about national-level multipliers.
Estimates can be used to form counterfactuals.
Identification is based on strong assumptions about the model structure and shocks processes.

6. A. Strengths and Weaknesses (continued)
Type
Advantages
Disadvantages
3. Subnational geographic cross-sectional or panel evidence
Identification is often much easier and stronger – uses applied micro identification.
Many different possible data sets.
Estimates only relative effects, so not directly informative about national multipliers (national factors are differenced out).
Requires additional identification assumptions to translate subnational to national multipliers, typically DSGE model for identification.
4. Individual industry, firm or household estimates of key parameters (such as MPC).
Identification is often much easier and stronger – uses applied micro identification.
Many different possible data sets.
Only estimates some key micro parameters, so not directly informative about national multipliers.
A DSGE model is required to translate the micro parameter estimates to national predictions about multipliers.

7. B. Some Innovations and Lessons Learned
How to calculate multipliers in a dynamic environment.
The importance of anticipations and how to deal with them.
Improvements in fiscal shock identification.

8. How to calculate multipliers in a dynamic environment
A recent lesson learned from the literature is that an important source of the wide range of multiplier estimates is due to differences in the method for calculating multipliers.
I will highlight two commonly used methods that often lead to upward bias in multipliers.

9. Illustration for government purchases multipliers
Structural VAR (SVAR) using Blanchard-Perotti identification, which orders government spending first.
Quarterly data from 1939:1 – 2015:4.
5 variables:
- log real total government spending per capita
- log real GDP per capita
- log real federal tax receipts per capita
- 3-month Treasury bill interest rate
- inflation rate.
4 lags.

10. Estimated responses to government spending shock.
95-percent confidence bands

11. Calculating Multipliers
How do we use the dynamic responses of the log variables to multipliers to answer the question:
How much does GDP rise when government spending rises by $1?
I will show how seemingly small changes in the method can lead to large changes in the multiplier.

12. 1. Blanchard-Perotti (2002) Quasi-Multipliers
Compute the ratio of the log GDP response at horizon h to the impact response (i.e. horizon 0) log government spending.
Convert elasticities (since logs) to $ multipliers by multiplying the ratio in (a) by the sample average GDP/Gov (4.8 in this sample).

13. Blanchard-Perotti quasi-multipliers
Blanchard and Perotti’s quasi-multipliers are above 2 at the peak.

14. Mountford-Uhlig Method
Compute the ratio of the present value of the cumulative responses.
Convert elasticities to $ multipliers, as with previous method.

15. Blanchard-Perotti quasi-multipliers
Mountford-Uhlig cumulative multipliers

16. Elasticities versus Multipliers
The standard log specification yields estimates of elasticities, not multipliers.
The standard method has been to convert elasticities to multipliers by multiplying by the sample average of Y/G.
𝑑𝑌 𝑑𝐺 = 𝑑𝑙𝑛(𝑌) 𝑑𝑙𝑛(𝐺) ∙ 𝑌 𝐺
Alternative method: Transform the variables to same units before estimation. Hall, Barro-Redlick divided changes in Y and G by lagged Y.
Gordon-Krenn: 𝑌 𝑡 𝑌 𝑡 𝑃 , 𝐺 𝑡 𝑌 𝑡 𝑃

17. Blanchard-Perotti quasi-multipliers
Mountford-Uhlig cumulative multipliers
Cumulative,
Gordon-Krenn transformation

18. State dependent biases
Sims and Wolff (2018) show that the conversion factor also
tends to bias multipliers differentially, making them seem much higher during recessions.
The intuition is simple: because GDP is cyclical but government spending is not, 𝑌 𝑡 / 𝐺 𝑡 is procylical. However, the practice of using a sample average of 𝑌/𝐺 to convert elasticities to multipliers makes the multipliers appear more countercyclical than they really are.

19. Additional issue for tax multipliers
Most tax multipliers reported are based on the legislative forecast of the budget impact, not taking into account dynamic feedback.
But tax cuts raise GDP, which mitigates the negative effect of the tax cut on tax revenue.
Tax multipliers are even greater (in magnitude) if reported relative to the actual change in tax revenue.

20. Bottom Line
Many of the big differences in reported multipliers are not due to the estimation method, sample, etc.
Rather, they are due to the reporting of quasi-multipliers that don’t take into account the dynamic path of government spending or to the use of ad hoc conversion factors to deal with estimates based on logs.

21. Other innovations
The importance of anticipations and how to deal with them.
econometrics of fiscal foresight
measures of news – narrative, bond spreads, professional forecasts.
Improvements in fiscal shock identification.
- narrative series
- increased understanding of sensitivity of SVARs

22. 2. Current Range of Leading Estimates of Fiscal Multipliers
Scope of summary:
Multipliers within the first two to five years.
Industrialized countries.
Estimates based on a variety of methodologies: time series models, narratives, and estimated New Keynesian DSGE models.
Excludes estimates that do not use the best practices.
Ranges shown are for the majority of the estimates, but don’t include some notable outliers produced with good methodology.

23. Multipliers on Government Purchases
ZLB or monetary accommodation
variety of methods
Recession/slack
0.6 to 1
1.5 to 2
Talk about which ones are more or less certain.
Time series and DSGE
aggregate data, many countries
general government purchases
sample averages
cross-section, panel
subnational
general government purchases, transfers

24. Multipliers on Taxes and Transfers
Temporary tax rebates
Household-level data, estimated MPCs
tax rate changes
Estimated DSGE models
Say I will explain
tax rate changes
Time series, narrative identification
aggregate data, many countries
temporary transfers, tax rebates
narrative identification, case studies, DSGE
aggregate data

25. Summary of Aggregate Evidence
Most multipliers on government consumption spending estimated with linear models on national data are below 1. These apply to both the BP and defense spending estimates and DSGE estimates.
The tax change multipliers are estimated to be between -2.5 and -3 across numerous countries, but less than 1 in DSGE models.
The fiscal consolidation multipliers depend on whether the consolidation was mostly through spending or taxes, with the effects greater in magnitude for tax based consolidations.
Public investment multipliers could be higher: Iltzeki, Mendoza, Vegh (2013) find 1.5 to 1.6.

26. 3. What do household and cross-sectional estimates tell us about national multipliers?

27. Motivation
The household and cross-sectional estimates are often considered to be on firmer ground because:
The experiments are often cleaner.
Identification is often stronger.
There are multiple data sets that can be used.
Some recent work on NK models specifically analyzes how cross-sectional multipliers translate to national multipliers.
Nakamura-Steinsson (2014), Farhi-Werning (2016), Chodorow-Reich (forthcoming).
In fact, Farhi-Werning & Chodorow-Reich argue that the cross-state multipliers for outside financed transfers are lower bounds on the national multiplier at the ZLB.
Moreover, a number of TANK and HANK models are calibrated to the estimates of high MPCs out of temporary tax rebates found by Parker and others.

28. Outline of this Section
Cross-state ARRA estimates.
I will question the plausibility of using these estimates at the national level.
Empirical – do the estimates answer the right question?
Theoretical - I will consider the theoretical assumptions that suggest they are a lower bound on the national multipliers.
2. Estimates of MPCs at the household level.
- I will first review the leading recent studies on the MPCs out of temporary tax rebates.
- I will then argue that the aggregate implications are implausible.
- I will offer several possible resolutions.

29. Cross-State Estimates of ARRA Multipliers
Chodorow-Reich (forthcoming AEJ: Economic Policy) uses three leading instruments from the literature and standardizes the multiplier estimates obtained from cross-state differences in the ARRA.
Across all three instruments, as well as combined, he finds that each $100K led to approximately 2 job-years, i.e., creating one job year costs approximately $50K in stimulus funds.
His employment estimates imply output multipliers between 1.7 and 2.
Using a theoretical NK model, he concludes that these cross-state estimates are lower bounds on the national estimates.

30. More Details on the Chodorow-Reich Estimates
Chodorow-Reich (C-R) multiplier estimate is based on the estimates the cumulative job-years created for 24 months after the ARRA.
Month 0 is Dec Stimulus was passed in Feb. 2008, but was anticipated.
He does not base his conclusions on estimates past Dec because another stimulus was passed and it may confound the estimates.
As of Dec. 2010, almost $654 billion of the stimulus had been spent.

31. More Details on the Chodorow-Reich Estimates
In the online appendix, he shows the actual impulse response of employment to the initial ARRA shock.

32. Plausibility: A Counterfactual
Chodorow-Reich claims that his estimates are a lower bound on the national estimates.
To check the plausibility, I use Chodorow-Reich’s impulse response estimates to calculate the following counterfactual:
What would the unemployment rate have been had there been no ARRA? This is a partial equilibrium analysis, which ignores general equilibrium effects.
I use the $654 billion of stimulus shock together with his IRF estimates to calculate the implied additional employment, and then use it to find the implied unemployment rate.

33. Counterfactual
Counterfactual if no ARRA
Actual unemployment rate
The counterfactual estimates are based on Chodorow-Reich (2017) estimates of the effects of the ARRA on employment by month through December 2010.

34. Explorations for Possible Explanations
Issues with Chodorow-Reich’s estimates?
Is there a problem with the second step identification: the theory that says that the cross-state estimates should have been a lower bound on the national multiplier during the ZLB?

35. 1. Re-examining Chodorow-Reich’s Estimates
95% confidence band:
(0.85, 3.17)
Relevant instruments
Passes over-identification test

36. Why the Cross-State ARRA multipliers don’t Aggregate
They are not nationally representative.
- The ARRA studies use per capita variables by state but don’t weight their regressions by state population, i.e., they give North Dakota the same weight as California.
- If treatment effects are heterogeneous across states, then the unweighted estimates won’t be nationally representative.
They don’t account for all government spending.
Much of the ARRA consisted of federal transfers to states. Several studies have found that induced state spending was more than one-for-one (e.g. Leduc and Wilson (2017)).

37. What happens if we correct the estimates?
Using Chodorow-Reich’s replication files, I re-estimate his model but weight each state by population and use total state and local induced spending.
The estimates are for job-years per $100K but that is approximately equal to the output multiplier.
Chodorow-Reich estimate
Weighted by population
All govt spending, weighted by pop
Multiplier
2.01
1.15
0.89
Robust s.e.
(0.59)
(0.72)
(0.45)
Bottom line: now the ARRA multiplier estimates look like the average historical aggregate estimates.

38. 2. Second Step of Identification: The Theory
Chodorow-Reich using a lower bound argument based on analysis by Farhi-Werning (2016), p

39. 2. Second Step of Identification: The Theory
The key comparison is to national multipliers at the ZLB. The federal funds rates was at the ZLB during the ARRA, so C-R argues that the national multipliers on the ARRA were greater than the estimated state multipliers.
But there are several reasons to believe that the theory for national multipliers at the ZLB doesn’t apply during the ARRA.
Swanson and Williams (2014) argue that longer term yields matter and they show that “1- and 2-year Treasury yields were surprisingly unconstrained throughout 2008 to 2010, suggesting that monetary and fiscal policy were about as effective as usual during this period.” (abstract)
The key mechanism for greater fiscal multipliers during a ZLB is the expected inflation channel (leading to lower real interest rates). However, Dupor and Li (2014) test for this channel during the recovery period using surveys of professional forecasters and other data and find that the expected inflation effect is too small to lead to large multipliers.

40. 2. Household-Level MPC Estimates
MPC estimates from temporary tax rebates are based on some of the most impeccable applied-micro type empirical work.
Johnson, Parker, Souleles (JPS) (AER 2006) studied the 2001 tax rebate and Parker, Souleles, Johnson, and McClelland (PSJM) (AER 2013), Broda-Parker (JME 2014) studied the 2008 tax rebate. In each case:
tax rebates distributed to households over several months, with timing randomized by the last two digits of Social Security numbers.
the authors added special questions to surveys (CE for JPS, PSJM; Nielsen Survey for Broda-Parker) that measured the time, amount, and form of the rebate for each household.
I will focus on the PSJM study of 2008 rebate.

41. PSJM Specification What does the estimate of β2 tell us?
Controls: age and family size
The economic stimulus payment received by household in t+1.
Month dummy variables for every month in the sample.
The change in household i expenditures from t to t+1 (quarterly)
What does the estimate of β2 tell us?
It tells us how much consumption rises in t+1 per $ ESP received in t+1 relative to:
Household i’s consumption changes in other months.
The average change in consumption in month t+1 for all households in the sample.

42. Specifications Extensions of the basic specification
Use dummy for receipt of the ESP as an instrument for the dollar value of ESP and obtain very similar estimates.
Allow for lagged effects as well, adding ESPi,t in the regression.

43. PSJM 2008 Tax Rebate Results
Tax change enacted in Feb $100 billion in rebates distributed from May through July 2008.
130 million households received these one-time rebates, $300 - $600 for singles, $600 - $1200 for married couples, plus $300 per child.
PSJM “We find that on average households spent about 12 to 30 percent of their stimulus payments … on nondurable consumption goods and services during the three-month period in which the payments were received. This response is statistically and economically significant. We also find a significant effect on the purchase of durable goods and related services, primarily the purchase of vehicles, bringing the average response of total CE consumption expenditures to about 50 to 90 percent of the payments during the three-month period of receipt.” (p. 2531)

44. Are the Aggregate Implications Plausible?
In 2008 and 2009, Feldstein (2008) and Taylor (2009) wrote commentary based on aggregate data, arguing that most of the 2008 tax rebate was saved.
Shapiro and Slemrod (2009) and Sahm, Shapiro, and Slemrod (2012) found smaller MPCs (0.13 to 0.25) using the Michigan Survey based on questions about individuals’ intent. They also made the point that the aggregate data did not seem to be consistent with the high MPCs estimated by Parker and co-authors.

45. Let’s consider some counterfactuals:
What would have happened to aggregate consumption and saving had there been no 2008 tax rebate?

46. 1. Sahm-Shapiro-Slemrod (2012) Counterfactual
When comparing their smaller estimated MPCs to PSJM’s estimates, they conduct a counterfactual based on PSJM’s motor vehicle estimates.
Recall that purchases of motor vehicles was the prime reason PSJM found such a high MPC.
At the back of the Sahm-Shapiro-Slemrod paper, Table 14 uses PSJM’s estimated MPC on new motor vehicles to calculate the partial equilibrium counterfactual for aggregate expenditures on motor vehicles if there had been no tax rebate.

47. 1. Sahm-Shapiro-Slemrod (2012) Counterfactual
Actual expenditures
Counterfactual if no tax rebate
Sahm-Shapiro-Slemrod (2012) Table 14, converted to monthly rates, updated with revised actual expenditures. Based on PSJM headline MPC estimate of on new vehicles. SSS assume it is spread evenly over current subsequent month.

48. 2. Saving Rate Counterfactual
Based on an argument made by Feldstein (2008), Taylor (2009), and Slemrod-Shapiro (2009).
I will show using aggregate data:
how big the rebate was relative to disposable income
what happened to the saving rate
a back-of-the-envelope calculation for the implied MPC

49. 2. Saving Rate Counterfactual
Rebate rate = 100∙ 𝑅𝑒𝑏𝑎𝑡𝑒 𝑁𝑜𝑛−𝑅𝑒𝑏𝑎𝑡𝑒 𝐷𝑖𝑠𝑝𝑜𝑠𝑎𝑏𝑙𝑒 𝐼𝑛𝑐𝑜𝑚𝑒
Saving rate = 100∙ 𝑆𝑎𝑣𝑖𝑛𝑔 𝑁𝑜𝑛−𝑅𝑒𝑏𝑎𝑡𝑒 𝐷𝑖𝑠𝑝𝑜𝑠𝑎𝑏𝑙𝑒 𝐼𝑛𝑐𝑜𝑚𝑒

50. as a % of Non-Rebate Disposable Income
2008 Rebate and Saving Rate
as a % of Non-Rebate Disposable Income
Saving rate
Rebate
The rebate peaked above 5% of disposable income.
Saving rate was relatively constant the first four months of 2008, then rose significantly, then came down to earlier levels in 2008m8, then rose once Lehman Brothers failed.

51. Back of the Envelope Calculation
Rebate rate = 100∙ 𝑅𝑒𝑏𝑎𝑡𝑒 𝑁𝑜𝑛−𝑅𝑒𝑏𝑎𝑡𝑒 𝐷𝑖𝑠𝑝𝑜𝑠𝑎𝑏𝑙𝑒 𝐼𝑛𝑐𝑜𝑚𝑒
Saving rate = 100∙ 𝑆𝑎𝑣𝑖𝑛𝑔 𝑁𝑜𝑛−𝑅𝑒𝑏𝑎𝑡𝑒 𝐷𝑖𝑠𝑝𝑜𝑠𝑎𝑏𝑙𝑒 𝐼𝑛𝑐𝑜𝑚𝑒
Considering only the partial equilibrium, rewrite saving:
Suppose we assume the saving rate from DPIx was unchanged before Lehman Brothers.
Saving rate =
100∙ 𝑆𝑎𝑣𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝑛𝑜𝑛−𝑅𝑒𝑏𝑎𝑡𝑒 𝐷𝑖𝑠𝑝. 𝐼𝑛𝑐 + 1−𝑚𝑝𝑐 ∗𝑟𝑒𝑏𝑎𝑡𝑒 𝑁𝑜𝑛−𝑅𝑒𝑏𝑎𝑡𝑒 𝐷𝑖𝑠𝑝𝑜𝑠𝑎𝑏𝑙𝑒 𝐼𝑛𝑐𝑜𝑚𝑒

52. as a % of Non-Rebate Disposable Income
2008 Rebate and Saving Rate
as a % of Non-Rebate Disposable Income
Rebate
Saving rate less Jan. value
Area reveals MPC
I renormalized the saving rate so January value is 0, so that we can compare increase to the rebate percent.
The area between the rebate percent and the saving rate percent from 2008m4 should be approximately equal to cumulative MPC. I calculate 0.13.

53. Implications of the Counterfactuals
Applying the micro estimates to aggregate data leads to implausible counterfactuals:
PSJM estimates imply a cumulative MPC of 50 to 90%, with much of it driven by transportation (mostly motor vehicles). Sahm-Shapiro-Slemrod’s counterfactual for new purchases of motor vehicles shows an implausible counterfactual.
My back-of-the-envelope calculation based on aggregate saving rates suggests an MPC of about 13%

54. Possible Explanations
Precision. Some of the micro estimates are not very precise. In PSJM’s contemporaneous model, the estimated MPC for total consumption is with a standard error of In the cumulative model, the estimate is with a standard error
Micro MPCs ≠ Macro MPCs. Recall that the household level estimates give only relative MPCs, relative to the household’s MPC in other months and relative to the aggregate during the month. If households are waiting until the rebate arrives to put a down payment on a car, their spending may fall in the non-rebate months.

55. 2. Micro MPCs ≠ Macro MPCs
PSJM’s estimates of β2 are significantly different from 0, so they can reject the pure LCPIH with rational expectations at standard significance levels.
However, I will argue that they cannot interpret their estimate of β2 as an MPC that is relevant for the aggregate.
I will simulate two simple models of consumer behavior that yield their β2 but that do not apply to aggregate consumption expenditures.

56. Simple Simulations Balanced panel of 6,000 households over 12 months.
All households have mean monthly disposable income of 100 (without rebate) and they save 4% on average (which matches pre-rebate saving rate).
Both income and consumption are hit by iid normal shocks, with variance 1 respectively.
4,000 households receive a one-time tax rebate, equal to 20% of one month’s income, announced in month 1 is distributed randomly, calibrated to the monthly arrival rate in the actual data and key statistics in Parker et al.

57. A. Hand-to-Mouth Consumers Experiment
λ of the households are hand-to-mouth consumers, with MPC = 1, (1- λ) are LCPIH consumers who spend 4% of the rebate each year.
Interestingly, there is not a one-to-one mapping of λ to the estimated β2 . In order to estimate a β2 = 0.51 on the household data I need a λ = 0.43.
When I aggregate the data (not allowing for any general equilibrium effects) and plot the same rebate percent of disposable income and saving rate as a percent of disposable income (both excluding rebate in the denominator), I obtain the following graph:

58. A. Hand-to-Mouth Consumers Experiment
Simulated Data
Estimated β = 0.51 in the household data.
Actual Data
Thus, the fraction hand-to-mouth consumers that produces the PSJM estimated β in the micro data cannot produce the aggregate data.

59. B. Long-run LCPIH with Short-run Bunching Experiment
All households are assumed to be permanent income consumers on an annual basis, spending a cumulative annuity rate of 5% of the rebate spread over the 12 months.
However, because of short-term liquidity constraints, wedges between borrowing and lending rates, discrete purchases, down payments on cars, etc. they intertemporally substitute their spending to the month in which the rebate check arrives.
Timing:
Month relative to rebate check arrival
% spent of rebate -2
- 3 -1
-18 25
Note that the concurrent MPC = 0.25.

60. B. Long-run LCPIH with Short-run Bunching Experiment
When I estimate the PSJM equation on my household panel data, I estimate β2 = (with a standard error of 0.002).
When I aggregate the data (not allowing for any general equilibrium effects) and plot the same rebate percent of disposable income and saving rate as a percent of disposable income (both excluding rebate in the denominator), I obtain the following graph:
This looks much closer to the actual aggregate data.
Rebate
Saving rate

61. Bottom Line:
Most micro studies are estimating relative MPCs, which are not suitable for macro uses.

62. Lessons Learned about Micro Estimates of MPCs
MPCs estimated at the household level usually reveal only relative effects and cannot be used to draw aggregate conclusions. For example, a high contemporaneous effect of a temporary rebate on consumption growth is consistent both with hand-to-mouth consumers and households intertemporally substituting expenditures because of short-term liquidity constraints.
Counter-factual aggregate calculations are a helpful adjunct to judge the plausibility of household estimates for macro.

63. My conclusions about household and cross-state estimates
There is no “applied micro free lunch” for macroeconomists.
The household MPC and cross-state multiplier parameter estimates are easier to identify and estimate precisely because they are micro parameters, not macro parameters!
There is a wide chasm between those estimates and the ones we need for macroeconomics.
Theory can be helpful, but the assumptions of the theory are just an additional set of identifying assumptions.

64. Conclusions There really has been a renaissance in fiscal research.
We know much more now than we did ten years ago.
Garden-variety government purchases multipliers are probably between 0.6 and 1, though there are a few credible estimates above 1.
Tax rate change multipliers are probably between -2 and -3.
Multipliers on infrastructure and during periods of monetary accommodation are probably above one, possibly substantially above one, but more research should be done to assess the robustness of these results.

66. 4. Were Multipliers Greater in the Wake of the Financial Crisis?

67. European Fiscal Consolidations
In the wake of the financial crisis, numerous European countries undertook fiscal consolidations.
Some were more spending-based and others were more tax-based.
Studies using narrative evidence for identification find that on average tax-based consolidations have much larger output effects than spending-based consolidations.
These results are consistent with the U.S. findings for tax versus spending multipliers.

68. Alesina, Favero, and Giavazzi (2018, forthcoming) find that once they control for tax vs. spending features, multipliers in the wake of the financial crisis are not different from those on average.
Blanchard and Leigh (2013, 2014) present indirect evidence of multipliers larger than those assumed initially in large macro models, using correlations between GDP forecasts errors and the size of the fiscal consolidations.
Górnicka et al. (2018) gather data on the assumed multipliers and find that they were They conclude that the ex post “true multipliers” were higher but still less than 1.