# RDP 2020-07: How Many Jobs Did JobKeeper Keep? Appendix B: Additional Details on the Model

Equation (2) gives the effect of JobKeeper worker eligibility on employment, which is different to the effect of actually receiving JobKeeper on employment. In Section 6.2 we mentioned that we can obtain an estimate of the latter by applying a scaling factor to the former. In this appendix we discuss this scaling approach in detail. To do this, we start by describing the ‘ideal model’ that we wish that we could estimate, but are unable to given the limitations of the data we have access to.

## B.1 The Ideal Model

Ideally we would directly estimate the effect of receiving JobKeeper on the probability of being employed in month j, for those people who were employed on 1 March 2020 (the eligibility date for JobKeeper),

(B1) $E i,j =p+τJobKeepe r i +φΔRevenu e i,f,j + ω i,j$

where JobKeeperi equals one if worker i was participating in the JobKeeper program at time j, and zero otherwise. A key control is $\text{Δ}$Revenuei,f,j which measures the COVID-19-related revenue loss experienced by the firm f that employed worker i on 1 March 2010 (in cumulative terms to month j). The parameter of interest is $\tau$ , which is the effect of JobKeeper on the probability of being employed in period j, controlling for the revenue shock that the firm experienced during the crisis.

The first thing that prevents us from estimating the ideal model is that we do not observe firm revenues in our dataset. If this were the only missing variable, we could feasibly estimate the following model,

(B2) $E i,j =ρ+τJobKeepe r i + ψ i,j$

However, if we leave $\text{Δ}$Revenuei,f,j out of the model as in Equation (B2), the estimate of $\stackrel{^}{\tau }$ would be biased, because receiving JobKeeper is correlated with firm revenue losses due to the firm-eligibility test. Formally, $\mathrm{cov}\left(JobKee{p}_{i},{\psi }_{i,j}\right)=\phi \mathrm{cov}\left(JobKeepe{r}_{i},\text{\hspace{0.17em}}\text{Δ}Revenu{e}_{i,f,j}\right)\ne 0.$

## B.2 A Feasible Model

We can overcome the omitted variable bias in Equation (B2) by instrumenting for JobKeeperi and focusing on casuals with 6–23 months of tenure. An obvious instrument in this case is our dummy variable for whether the worker passed the 12-month tenure test for casuals, Eligi. This instrument is relevant (positively correlated with receiving JobKeeper) and exogenous (unrelated to $\text{Δ}$Revenuei,f,j). Our decision to focus on a narrow range of job tenure can be motivated using this exogeneity assumption. If we included a broader range of tenure it would raise the likelihood of there being some unobservable factor correlated with both the likelihood of receiving JobKeeper and firm revenue losses.

In addition to accounting for omitted variables, this instrumental variables (IV) approach also gives us a way of estimating the parameter that policymakers are most interested in – namely, the effect of JobKeeper on workers who received JobKeeper. This is the case even though we do not have a direct measure of whether a person received JobKeeper in our dataset. To see this, note that the set-up with a binary treatment and binary instrument lends itself to the Wald IV estimator:

$τ IV = cov( E i,j , Eli g i ) cov( JobKeepe r i , Eli g i ) = E( E i,j | Eli g i =1 )−E( E i,j | Eli g i =0 ) E( JobKeepe r i | Eli g i =1 )−E( JobKeepe r i | Eli g i =0 )$

Casual employees cannot access JobKeeper if they do not satisfy the tenure test, so this simplifies to:

(B3) $τ IV = E( E i,j | Eli g i =1 )−E( E i,j | Eli g i =0 ) E( JobKeepe r i | Eli g i =1 )$

The numerator of Equation (B3) is the effect of JobKeeper worker eligibility on employment. This is an intent-to-treat effect. We can estimate this effect using the LFS micro data since we observe information on whether the worker is employed on a casual basis and whether they have been engaged long enough to satisfy the 12-month tenure rule. Our estimate of the numerator in Equation (B3) is our estimate of $\stackrel{^}{\delta }$ from using OLS on Equation (2) (a linear probability model).

The denominator of Equation (B3) is the probability of receiving JobKeeper, conditional on being worker-eligible. This quantity cannot be estimated using the LFS micro data alone since we do not observe who receives JobKeeper. However, we can construct an estimate using a combination of ATO data and the LLFS:

(B4) $E( JobKeepe r i | Eli g i =1 )≈ total JobKeeper recipients no of workers satisfying worker eligibility ≈ 3.5m ∼10.26m ≈ 1 3$

Total JobKeeper recipients is from official sources based on administrative data (3.5 million over the April to May period according to Treasury (2020b)), while the total number of workers satisfying the worker-eligibility test is our estimate based on LFS micro data for February 2020. To obtain this estimate, we take the total number of employed people and subtract the number of casual employees with less than 12 months of job tenure. We also subtract people employed in the public sector or by a major bank, who are ineligible.

Our estimate of E(JobKeeperi|Eligi = 1) from Equation (B4) is imperfect since it pertains to a broader population than what we use to estimate the intent-to-treat effect in Equation (B3). The premise of the Wald estimator is that the numerator and denominator are drawn from the same population. The difficulty in doing this is that we do not receive sufficiently granular administrative data on the characteristics of individuals receiving JobKeeper. One dimension that we do have disaggregation in official JobKeeper numbers is by industry. As a robustness check, we examined whether our estimates were robust to stratifying the calculation of E(JobKeeperi|Eligi = 1) by industry, which accounts for any differences in industry composition between our estimation sample for Equation (B3) and the broader population of worker-eligible individuals. This industry stratification had no material effect on our results; our estimate of E(JobKeeperi|Eligi = 1) is still close to ⅓.

The above discussion suggests that to obtain an estimate of the causal effect of JobKeeper on employment we can simply estimate Equation (2) using OLS and then divide the estimate of $\stackrel{^}{\delta }$ by a scaling factor of ⅓,

(B5) $τ ^ IV = δ ^ 1 3$

We do this calculation in Section 6.2. We calculate standard errors for ${\tau }_{IV}$ under the simplifying assumption that $\delta$ is random but E(JobKeeperi|Eligi = 1) is known. This does not account for the sampling variance of the scaling factor, which means our standard errors for ${\stackrel{^}{\tau }}_{IV}$ are likely to be too small.

Our estimate of ${\tau }_{IV}$ yields the average treatment effect on the treated (ATT). That is, it tells us the average causal effect of receiving JobKeeper on employment for those who actually received the subsidy. For this reason, our estimates only pertain to the effects of JobKeeper on people working at eligible firms.

## B.3 Implied Aggregate Effect

To estimate the implied effect of JobKeeper on aggregate employment in Section 8.1 we use the following formula,

Total employment effect $={\stackrel{^}{\tau }}_{IV}×3.5\text{m}$

Or, equivalently

Total employment effect $=\stackrel{^}{\delta }×10.26\text{m}$

Autor et al (2020) use a similar calculation to estimate the aggregate effect of the PPP on US employment.

## Footnotes

In this simplified model, the revenue measure excludes any revenue from the JobKeeper subsidy itself. There are likely other worker- and firm-level variables we need to control for in Model (B1) to get an unbiased estimate of $\tau$ , but for expositional simplicity we assume that all relevant confounders are captured by the $\text{Δ}$Revenuei,f,j variable. 

We do not adjust for visa status because we did not exclude temporary visa holders from the estimation sample for Equation (2). 

The notion that the numerator and denominator of Equation (B3) do not need to come from the same sample is based on the two-sample IV (TSIV) estimator (see Angrist and Pischke (2009, pp 147–148)). 

E(JobKeeperi|Eligi = 1) $={\sum }_{k=1}^{19}\left[\left(\frac{total\text{\hspace{0.17em}}JobKeeper\text{\hspace{0.17em}}recipient{s}_{k}}{no\text{\hspace{0.17em}}of\text{\hspace{0.17em}}workers\text{\hspace{0.17em}}satisfying\text{\hspace{0.17em}}worker\text{\hspace{0.17em}}eligibilit{y}_{k}}\right)×\frac{estimation\text{\hspace{0.17em}}sampl{e}_{k}}{total\text{\hspace{0.17em}}estimation\text{\hspace{0.17em}}sampl{e}_{k}}\right]$, where k denotes 1-digit ANZSIC 2006 industry division. The number of JobKeeper recipients by industry is from Treasury (2020b, p 43), which we adjust (proportionally) so that industry totals sum to 3.5 million. 

Our IV approach yields a local average treatment effect (LATE) – that is, the effect of JobKeeper on employment for those whose treatment status (receiving JobKeeper) is affected by worker eligibility. Always-takers do not exist in our set-up because a worker could not receive JobKeeper if they were not worker-eligible. For this reason, the LATE also corresponds to the ATT (Angrist and Pischke 2009, p 160).