## 5.1 Regression Specification / 7.3 Crossers and Non-Crossers

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### Regression identification and specification
- Outcome and setup:
  - Outcome g_it: growth rate in turnover (or cost) of firm i at time t; ε_it clustered at firm level.
  - Treatment variables: A_j (dummies for bins above threshold, width £2,000), B_j (dummies for bins below threshold, width £2,000).
  - Window size W: number of bins above and below included.
- Controls and fixed effects:
  - Firm fixed effects f_i, year fixed effects τ_t, firm age fixed effects age_i,t.
  - Bin fixed effects b_i,k: 1 if firm i has turnover in bin k at t−1—allow growth to respond non-parametrically to lagged turnover.
- Identification assumptions:
  - (i) firms outside window W are not affected by threshold;
  - (ii) unobserved differences between firms with different turnover levels are stable over time.
- Data-driven choice of window:
  - Increase W until β_W is insignificant; this occurs at W = 10, implying reaction starts when turnover is about £20,000 below the threshold.
- Role of bin fixed effects:
  - Including bin fixed effects reduces magnitude of estimated β_j and α_j relative to specification without them.
  - Small catch-up effect just above threshold is insignificant once bin fixed effects included.

### Main reduced-form findings on slowdown near the VAT threshold
- Turnover growth:
  - Clear evidence of slowdown (β_j < 0) as firms approach VAT threshold.
  - Quantitative magnitude: effect is "up to 2 p.p."
  - Sample average growth: "around 8 p.p.", so slowdown is "a 25% slowdown" as firm nears threshold.
  - No evidence of accelerating growth once past threshold (α_j > 0 not found significant in regressions with firm fixed effects).
- Cost growth:
  - Similar pattern to turnover: input cost growth slows below the threshold with catch-up above the threshold in naïve plots; pattern remains when formal regressions are run.
- Robustness:
  - Re-running regression with turnover and cost deflated by fiscal year GDP deflator (baseline 100 in 2014-2015) yields very similar results; baseline uses nominal values.

### Mechanisms: real response versus reporting/evasion
- Two candidate mechanisms:
  - Real response: firms reduce scale of operations (reduced sales activity, business restructuring).
  - Evasion/reporting: under-report sales and/or over-report purchases so reported turnover stays below threshold.
- Empirical evidence favoring real adjustments:
  - Similarity of patterns for turnover and for input cost growth suggests coordinated changes in both sales and inputs rather than only under-reporting.
  - Author judgment: coordinated under-reporting across production chains and employees to keep reported turnover below threshold is "quite unlikely" in the UK because it "would require quite a lot of co-ordination."

### Crossers versus non-crossers: empirical patterns and interpretation
- Definitions:
  - Non-crossers: firms that never cross the VAT threshold during the sample.
  - Crossers: firms that start below and eventually expand above the VAT threshold.
- Key empirical facts:
  - HMRC/IPSOS survey: "Overall, 20% of unregistered borderline businesses admit to having taken some action to remain under the threshold and outside the VAT system."
  - In administrative data for incorporated firms, annual turnover growth starts to slow when firm turnover gets within about £20,000 of the threshold.
  - Growth slows by up to 2 percentage points when firms get close to the threshold; with the sample average growth rate of 8 percent, this implies a slowdown of up to 25 percent in the growth rate.
- Heterogeneous responses:
  - Non-voluntary registered firms slow strongly as they approach the threshold: growth falls 3-4 percentage points.
  - Voluntary registered firms do not slow down as they approach the threshold.
  - Flat Rate Scheme (FRS) uptake: 40% of the VAT registered firms in the sample are in the FRS; FRS firms slow down less before the VAT threshold than those not using the FRS.
- Crossers' observed behavior:
  - Crossers appear to show no slowdown in turnover or cost growth as they approach the threshold in observed data subsets.

### Selection effects explaining apparent absence of slowdown among crossers
- Two selection mechanisms:
  1. Exclusion of behavioral non-crossers: Non-crossers (about 20% of firms) who restrict growth below threshold are absent from the crossers sample, removing the group that drives pooled slowdown.
  2. Stochastic-selection effect from growth process:
     - Crossers below the threshold are enriched for firms that experienced positive shocks (growing faster than average).
     - Crossers above the threshold are enriched for firms that experienced negative shocks (growing slower than average).
     - These two effects can jointly offset a true slowdown around the threshold when attention is restricted to crossers.
- Simulation illustrating selection:
  - Simulated dataset parameters:
    - 100,000 firms with 10 years of turnover growth data.
    - Initial turnover drawn uniformly between £10,000 and £80,000.
    - VAT threshold set at £50,000.
    - Growth generated by: g_it = 10 − 0.1 y_{i,t−1} + 0.0005 y^2 − 2.5(y_{t−1}/50) I[y_{t−1} < 50] + ε_it, where ε_it ∼ N (0, 10).
  - Result: for chosen parameters, selection effects completely offset the built-in slowdown among crossers so crossers appear not to slow, despite a discontinuity in the whole-sample data.
- Empirical implication:
  - Studying only crossers can lead to substantial understatement of the true threshold effect; whole-sample analysis is necessary to capture aggregate slowdown.
  - It is not possible in the available data to perfectly separate firms that deliberately avoid crossing from firms that fail to cross due to unfavorable growth shocks.

### Heterogeneity: voluntary registration and Flat-Rate Scheme (FRS)
- Voluntary registration (VR):
  - VR_it = 1 if firm is below the threshold and registered in t and in t−1; NVR_it = 1 − VR_it.
  - Prevalence: 44 percent of firms below the threshold are voluntarily registered (stringent definition).
  - Diff-in-diff findings (Equation (2)):
    - Interaction of proximity bins with NVR_it shows NVR firms suppress growth more as they approach threshold (γ_j < 0 in OLS).
    - Instrumental-variable strategy: NV R_it instrumented by yearly average industry B2C ratio; first-stage F-statistic described as "very high".
    - IV estimates less precise but qualitatively similar to OLS.
- Flat-Rate Scheme (FRS):
  - Description: simplified VAT scheme where VAT liability equals single rate times total turnover (effectively a turnover tax); reduces record-keeping and compliance costs.
  - Eligibility/sample facts:
    - FRS introduced in 2002.
    - Eligibility threshold: fixed at £150,000 since 2011, £100,000 prior to that.
    - Exit threshold: turnover including VAT exceeds £230,000 since 2011.
    - Of registered firms in sample, 40% opted for the FRS.
  - Empirical findings (Equation (3)):
    - Interaction of proximity bins with FRS_it; instrument FRS_it with industry input cost ratio; first-stage F-statistic "very high".
    - Both OLS and IV estimations show FRS membership "appears to have an additional effect on mitigating slowdown."

### Counterfactual simulations: effect of alternative VAT thresholds on firm size
- Counterfactual construction:
  - Use bin fixed effects to compute counterfactual growth path for average firm starting at initial turnover £16,000 with α_j = β_j = 0; compute actual growth path using estimated α_j, β_j for different VAT thresholds; report ratio actual/counterfactual turnover.
- Findings:
  - Firms begin to slow when turnover enters window of £20,000 around threshold.
  - With a £40,000 threshold: after 10 years the firm is "nearly 8% smaller" than without threshold.
  - Simulation horizon limited to 27 years because sample maximum turnover approaches £200,000.
  - Trade-off in threshold setting:
    - Low threshold restricts growth when firm is small.
    - High threshold restricts growth when firm is large.
    - Long-run firm size is maximized at an intermediate threshold: £80,000 (in this framework).
  - Across thresholds considered (£40,000 to £120,000):
    - Transitional dynamics can make firm size up to 8% smaller than in the absence of the threshold.
    - Long-run fall in firm size is only 0.5-1.0% relative to no threshold.
    - Reduction in long-run firm size is minimised at £80,000, close to the current VAT threshold in the UK.

### Policy-relevant implications
- Behavioral margin and prevalence:
  - A salient, high VAT threshold induces a substantial behavioral margin: a non-trivial fraction of borderline firms take explicit actions to remain below the threshold (survey: 20% admit actions).
  - Specific actions reported in survey among those restricting turnover include:
    - Closing the business or stopping advertising: 47%.
    - Refusing or turning down work: 21%.
    - Asking customers to purchase materials: 16%.
    - Reducing prices to avoid reaching threshold: 10%.
    - Splitting the business or artificially separating products/services: 10%.
- Design features that mitigate growth suppression:
  - Simplified VAT schemes (FRS) and voluntary registration status materially affect firms’ responsiveness to size-based tax thresholds; these features can mitigate growth-suppressing effects.
- Trade-off in threshold setting:
  - Lowering the threshold constrains smaller firms earlier; raising it shifts constraints to larger firms.
  - Simulations suggest modest long-run effects on firm size (0.5-1.0%), but potentially larger transitional effects (up to 8% smaller during transitional dynamics).
- Identification and inference caveats:
  - Selection effects complicate inference for crossers; estimates on crossers likely understate true threshold effects.
  - Instrumental-variable strategies (industry B2C share for VR, industry input cost ratio for FRS) are used to address endogeneity; first-stage F-statistics reported as "very high" in these analyses.

*Source: wpiea2024033-print-pdf — Sections 5.1 and 7.3 (Small Firm Growth and the VAT Threshold: Evidence for the UK).*

### 5.1 Regression Specification ................................................................................... 19

### 5.1 Regression Specification

### Section index and immediate subsections
- 5.1 Regression Specification ................................................................................... 19
- 5.2 Main Results ................................................................................................ 21

### Related subsequent sections (listed in source)
- 6 The Effects of Different Thresholds on Firm Size 23
- 7 Firm Heterogeneity 24

### Subsections of Section 7 (listed in source)
- 7.1 Voluntarily Registered v.s. Non-Voluntary Registered Firms .................... 24
- 7.2 The Flat-Rate Scheme .......................................................................... 26

*Source: wpiea2024033-print-pdf - 5.1 Regression Specification ................................................................................... 19*

### 7.3 Crossers and Non-Crossers ............................................................................... 28

### 7.3 Crossers and Non-Crossers

### Key empirical findings on crossers vs non-crossers
- HMRC/IPSOS survey finding: "Overall, 20% of unregistered borderline businesses admit to having taken some action to remain under the threshold and outside the VAT system."
- Specific actions reported in the survey (as quoted in the source):
  - Closing the business or stopping advertising: 47% of those restricting turnover admitted this.
  - Refusing or turning down work: 21%.
  - Asking customers to purchase materials: 16%.
  - Reducing prices of products to ensure the VAT threshold is not reached: 10%.
  - Splitting the business by operating as a separate legal entity or artificially separating the business by product or service: 10%.
- In the administrative data for incorporated firms:
  - Annual turnover growth starts to slow when firm turnover gets within about £20,000 of the threshold.
  - Growth slows by up to 2 percentage points when firms get close to the threshold. With the sample average growth rate of 8 percent, this implies a slowdown of up to 25 percent in the growth rate.
- Heterogeneous responses:
  - Non-voluntary registered firms slow strongly as they approach the threshold: growth falls 3-4 percentage points.
  - Voluntary registered firms do not slow down as they approach the threshold.
  - Flat Rate Scheme (FRS) usage: 40% of the VAT registered firms in the sample are in the FRS; firms in the FRS slow down less before the VAT threshold than those that do not use the FRS.
- Crossers (firms that start below and eventually expand above the VAT threshold) appear to show no slowdown in turnover or cost growth as they approach the threshold in the observed data.

### Interpretation and mechanism
- The observed slowdown in pooled data is primarily driven by a group of "non-crossers" who deliberately remain outside the VAT system, rather than uniform temporary slowing among all firms.
- The similarity of patterns for turnover and for cost growth suggests responses are likely real adjustments (e.g., reduced sales activity or business restructuring) rather than solely tax-evasion reporting changes.
- The existence of voluntary registration and FRS uptake provides alternative equilibria: firms that have already internalized VAT compliance costs or chosen simplified taxation are less responsive to the threshold.

### Policy-relevant quantitative results and simulations
- UK VAT registration threshold context:
  - The UK threshold is high relative to OECD peers; the current registration threshold reported for 2019/20 is £85,000.
  - The study exploits variation in the threshold over time (examples of thresholds by year are reported in Table 1).
- Simulation results on the effect of different VAT thresholds:
  - Thresholds considered range from £40,000 to £120,000.
  - The size of the firm can be up to 8% smaller than in the absence of the threshold in transitional dynamics.
  - The long-run fall in firm size is only 0.5-1.0% relative to no threshold.
  - The reduction in long-run firm size (relative to no threshold) is minimised at £80,000, close to the current VAT threshold in the UK.

### Selection and inference caveats
- Selection effects complicate inference for crossers:
  - While crossers in the observed sample "appear" not to slow before crossing, selection into the crosser group implies that estimates for crossers are likely to understate the true effect of the threshold on firms that would otherwise have crossed.
- Identification strategy notes (from the broader analysis in the chapter):
  - The study distinguishes the effect of distance to the threshold from general turnover-related growth dynamics by dividing turnover into bins and using bin fixed effects.
  - Instrumental-variable strategies are used in heterogeneous-firm analyses: industry share of B2C sales is used to instrument voluntary registration (building on Liu et al. (2021)); industry input cost ratio is used in IV analysis of FRS take-up.

### Implications
- A salient, high VAT threshold can induce a substantial behavioral margin: a non-trivial fraction of borderline firms take explicit actions to remain below the threshold.
- Simplified VAT schemes (FRS) and voluntary registration status materially affect firms’ responsiveness to size-based tax thresholds; these design features can mitigate growth-suppressing effects.
- Policy trade-offs: lowering administrative burdens or adjusting threshold design could influence the prevalence of non-crossers and thus aggregate small-firm growth dynamics; simulations suggest a modest long-run effect on firm size but potentially larger transitional effects.

*Source: wpiea2024033-print-pdf — 7.3 Crossers and Non-Crossers (excerpted content from the provided PDF).*

### Section 7.3 below.

### Section 7.3

### Mechanisms of Growth Suppression Near the VAT Threshold
- Two potential mechanisms for observed slowdown in firm growth near the VAT threshold:
  - Real response: firms reduce growth in scale of operations.
  - Evasion: firms under-report sales and/or over-report purchases of inputs, reducing reported turnover and reported input purchases and labour cost growth proportionally.
- Evidence from Figure 3:
  - Panel (a): turnover growth by £1,000 bins in neighbourhood of normalized VAT threshold during 2004/05-2014/15 shows slowdown as threshold approached.
  - Panel (b): input cost growth exhibits a gradual slow-down approaching the threshold with catch-up above the threshold; qualitatively similar to turnover growth.
- Author judgment:
  - Coordinated under-reporting of both sales and inputs across production chains and employees to maintain reported turnover below the threshold is possible but "quite unlikely" in the UK because it "would require quite a lot of co-ordination."

### Regression Identification Strategy (Equation (1))
- Outcome g_it: growth rate in turnover (or cost) of firm i at time t; ε_it clustered at firm level.
- Treatment variables:
  - A_j: dummies for bins above threshold (width £2,000).
  - B_j: dummies for bins below threshold (width £2,000).
  - Window size W: number of bins above and below included.
- Controls:
  - Firm fixed effects f_i, year fixed effects τ_t, firm age fixed effects age_i,t.
  - Bin fixed effects b_i,k: 1 if firm i has turnover in bin k at t−1—allow growth to respond non-parametrically to lagged turnover.
- Identification assumptions:
  - (i) firms outside window W are not affected by threshold;
  - (ii) unobserved differences between firms with different turnover levels are stable over time.
- Data-driven choice of W:
  - Increase W until β_W is insignificant; this occurs at W = 10, implying reaction starts when turnover is about £20,000 below the threshold.
- Role of bin fixed effects:
  - Including bin fixed effects reduces magnitude of estimated β_j and α_j relative to specification without them.
  - Small catch-up effect just above threshold is insignificant once bin fixed effects included.

### Main Regression Results (Section 5.2)
- Bin width for main results doubled to £4,000, with W = 5 (five bins).
- Turnover growth:
  - Clear evidence of slowdown (β_j < 0) as firms approach VAT threshold.
  - Quantitative magnitude: effect is "up to 2 p.p."
  - Sample average growth: "around 8 p.p.", so slowdown is "a 25% slowdown" as firm nears threshold.
  - No evidence of accelerating growth once past threshold (α_j > 0 not found significant in regressions with firm fixed effects).
- Cost growth:
  - Similar pattern: cost growth slows below the threshold.
- Robustness:
  - Re-running regression with turnover and cost deflated by fiscal year GDP deflator (baseline 100 in 2014-2015) yields very similar results; baseline uses nominal values.

### Simulated Effects of Different VAT Thresholds on Firm Size (Section 6)
- Counterfactual construction:
  - Use bin fixed effects to compute counterfactual growth path for average firm starting at initial turnover £16,000 with α_j = β_j = 0.
  - Calculate actual growth path using estimated α_j, β_j for different VAT thresholds; report ratio actual/counterfactual turnover.
- Key simulation findings:
  - Firms begin to slow when turnover enters window of £20,000 around threshold.
  - With a £40,000 threshold: after 10 years the firm is "nearly 8% smaller" than without threshold.
  - Simulation horizon limited to 27 years because sample maximum turnover approaches £200,000.
  - Trade-off in threshold setting:
    - Low threshold restricts growth when firm is small.
    - High threshold restricts growth when firm is large.
    - Long-run firm size maximized at intermediate threshold: £80,000 (in this framework).

### Firm Heterogeneity (Section 7)
- Voluntary Registered (VR) vs Non-Voluntary Registered (NVR) firms (Section 7.1)
  - Definitions:
    - VR at time t: firm is currently below the threshold and registered, and was below threshold and registered in t−1.
    - NVR = not VR; by definition NV R_it = 1 − V R_it.
  - Prevalence:
    - In data, 44 percent of firms below the threshold are voluntarily registered (stringent definition).
    - Cross-country examples mentioned in text (not UK-specific): Australia 37 percent (of businesses fell below threshold while 92 percent of all business were registered for VAT); South Africa 50 percent of businesses not required to be registered were registered nevertheless.
  - Diff-in-diff regression (Equation (2)):
    - Allows interaction of proximity bins with NV R_it; omit firm fixed effects for precision; estimated only for firms below threshold.
    - Expectation: γ_j < 0 (firms with reason to stay below threshold suppress growth more).
    - Instrumental variable:
      - NV R_it instrumented by yearly average industry B2C ratio; first-stage F-statistic described as "very high".
    - Results:
      - OLS: all γ_j < 0 — NVR firms suppress growth more as they approach threshold.
      - β_j estimates become slightly positive for VR firms (VR firms appear not to slow down).
      - IV estimates less precise but qualitatively similar.

- Flat-Rate Scheme (FRS) heterogeneity (Section 7.2)
  - FRS description:
    - Simplified VAT scheme; VAT liability equals single rate times total turnover (effectively a turnover tax); reduces record-keeping and compliance costs.
    - FRS rates set by sector so average trader indifferent between FRS and regular VAT; firms with below-average input costs may benefit from FRS.
  - Eligibility and sample facts:
    - FRS introduced in 2002.
    - Eligibility threshold: fixed at £150,000 since 2011, was £100,000 prior to that.
    - Exit threshold: turnover including VAT exceeds £230,000 since 2011.
    - Many sample firms eligible given turnover cap at £200,000.
    - Of registered firms in sample, 40% opted for the FRS.
  - Regression (Equation (3)):
    - Interacts proximity bins with FRS_it; controls include age, year, and bin fixed effects.
    - Instrument FRS_it with industry input cost ratio; first-stage F-statistic "very high".
  - Results:
    - Both OLS and IV estimations show membership of FRS "appears to have an additional effect on mitigating slowdown."

*Source: wpiea2024033-print-pdf - Section 7.3*

### 7.3 Crossers and Non-Crossers

### 7.3 Crossers and Non-Crossers

### Distinction between firm types
- Two firm types are defined: firms that never cross the VAT threshold (non-crossers) and firms that eventually cross the VAT threshold (crossers).
- About 20% of firms are non-crossers who restrict their growth just below the threshold.

### Turnover growth of crossers and apparent absence of slowdown
- Figure 9 shows turnover growth for firms that cross the threshold (defined as firms whose lowest (highest) turnover was below the minimum (above the maximum) VAT threshold during the sample period).
- In that subset, there does not appear to be any slowdown in growth as firms approach the threshold, which contrasts with the sharp slowdown observed in the aggregate results (e.g., Figure 3).

### Two explanations for the discrepancy (selection effects)
- Explanation 1: Non-crossers (about 20% of firms) who restrict growth below the threshold are excluded from the crossers sample, so their slowdown is not observed in Figure 9.
- Explanation 2: Less obvious selection effect arising from the stochastic growth process:
  - Crossers include firms initially below the threshold that experienced a sequence of positive shocks in previous periods and therefore are growing faster than average below and near the threshold.
  - Crossers also include firms initially above the threshold that experienced a sequence of negative shocks in previous periods and therefore are growing slower than average above and near the threshold.
  - Combined, these two selection effects tend to increase observed growth below the threshold and reduce observed growth above the threshold among crossers, smoothing out any real slowdown.

### Simulation illustrating selection effects
- Simulated firm-year data-set parameters:
  - 100,000 firms with 10 years of turnover growth data.
  - Initial turnover for each firm drawn uniformly between £10,000 and £80,000.
  - VAT threshold set at £50,000.
  - Growth from year t to year t − 1, expressed in percentage terms, is g_it = 100 × ∆ln y_it and is generated by equation (4):
    - g_it = 10 − 0.1 y_{i,t−1} + 0.0005 y^2 − 2.5(y_{t−1}/50) I[y_{t−1} < 50] + ε_it
    - where y_{i,t−1} is lagged turnover, ε_it ∼ N (0, 10), and I[y_{t−1} < 50] is an indicator equal to 1 only when lagged turnover is below the threshold.
- Simulation results:
  - Panel (a) of Figure 10 (whole sample) shows a discontinuity at the threshold due to the slowdown built into (4).
  - Panel (b) disaggregates into three groups:
    1. Firms never above the threshold (sequence of negative growth draws).
    2. Firms always above the threshold (initially high turnover and sequence of positive growth draws).
    3. Firms that cross the threshold (both directions).
  - For the chosen parameter values, the two selection effects jointly completely offset the built-in slowdown, so there appears to be no slowdown in growth among crossers, consistent with the empirical Figure 9.

### Implications for empirical analysis
- The “true” data-generating process in the example includes a slowdown that shows up in the whole data-set but not when attention is restricted to crossers; therefore, studying all firms together is necessary to capture the slowdown.
- The analysis is complicated by the presence of firms that may intentionally avoid crossing the threshold (behavioral non-crossers). Ideally these firms would be separated and studied individually.
- Practical limitation: It is not possible to distinguish in the data between firms that deliberately avoid crossing the threshold and firms that intend to cross but experience slow or negative growth during the sample period; this prevents clean separation of behavioral non-crossers.

*Source: 7.3 Crossers and Non-Crossers, Small Firm Growth and the VAT Threshold: Evidence for the UK (Working Paper No. WP/2024/033).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024033-print-pdf.pdf_