Problem: Stata DID Regression and Result Functions

The problem is to write Stata functions for DID-related analyses, including loading data, running multiple reghdfe models with fixed effects and clustered SEs, extracting key results (N, R², coefficients, SE, p-values), and displaying results clearly to replicate specified functionality.


Input/Output: The problem takes a Stata DTA file (e.g., data_verification.dta) containing variables IN, IA, RDF, RDP, DID, DID_lag1, KDM, c (cross-sectional unit), y (time), iy (industry-year), provy (cluster variable), and a global variable $x with control variables (Size2011y, Capital2011y, Aslbrt2011y, ROA2011y, Opicrt2011y, Cal2011y, LrgHldRt2011y, Economic2011y, Industry2011y, PC). It generates four Stata functions: did_analysis (loads the DTA, sets xt structure, computes pairwise correlations for IN_city, IA_city, DID, runs a lagged DID reghdfe regression, extracts and displays N, R², DID coefficient, SE, p-value), did_regressions_display (runs three reghdfe models, extracts/displays stats), did_analysis_triple (runs three reghdfe models including an interaction term, extracts/displays stats), reg_with_interaction_report (runs an interaction model, extracts/displays stats). All models use absorb(c y iy) and vce(cluster provy). Test cases verify outputs (e.g., N1=17540, r2_1=0.6782) against expected values with a 0.0001 tolerance.

Required dependencies:

Sub-steps:
step_number: 1
step_description_prompt: Please write Stata code to create a function named did_in_analysis
2. Function Definition: A Stata analysis workflow that implements panel data configuration, pairwise correlation analysis, fixed-effects regression, and regression result extraction with formatted display to evaluate the relationship between the lagged DID variable and IN variable
3. Algorithm Flow:
   - Step 1: Load the dataset from "../test_data/data_verification.dta" and clear existing data in memory
   - Step 2: Configure panel data structure using xtset with cross-sectional identifier 'c' and time identifier 'y'
   - Step 3: Define a global macro named 'x' containing the variables: Size2011y Capital2011y Aslbrt2011y ROA2011y Opicrt2011y Cal2011y LrgHldRt2011y Economic2011y Industry2011y PC
   - Step 4: Execute pairwise correlation analysis using the pwcorr_a command on variables IN_city, IA_city, DID, with significance star thresholds set as star1(0.01), star5(0.05), star10(0.1)
   - Step 5: Run a fixed-effects regression using reghdfe, regressing IN on l.DID, absorbing fixed effects for 'c', 'y', 'iy', and using cluster-robust standard errors clustered by 'provy'
   - Step 6: Extract regression statistics: sample size from e(N), R-squared from e(r2), coefficient of l.DID from _b[l.DID], standard error from _se[l.DID]. Calculate t-value as the ratio of l.DID coefficient to its standard error, p-value as 2*ttail(e(df_r), abs(t-value))
   - Step 7: Display the regression formula "IN ~ l.DID ", followed by the sample size, R-squared, and the coefficient in the format "[beta] (se=[se], p=[p])"
4. Mathematical Principles:
   - Pairwise Correlation: Calculate Pearson's correlation coefficients between the specified variables, test significance with t-tests, and mark coefficients with stars based on p-value thresholds: *** for p < 0.01, ** for p < 0.05, * for p < 0.1
   - Fixed-Effects Regression: Estimate the linear model: IN = β*l.DID + α_c + α_y + α_iy + ε, where α_c represents cross-sectional fixed effects, α_y represents time fixed effects, α_iy represents industry-time fixed effects. Cluster-robust standard errors are used to account for within-cluster correlation at the 'provy' level. The p-value is computed using the two-tailed t-distribution with residual degrees of freedom, as 2 * ttail(df_r, |t|)
5. Technical Requirements:
   - Utilize the exact specified Stata commands: xtset, global, pwcorr_a, reghdfe, display, ttail
   - Extract regression results strictly from Stata's built-in e() return values and compute t and p values manually as defined
   - Adhere to standard Stata syntax structure; do not use program define
6. Notes:
   - Do not add any functionalities beyond the specified workflow
   - Ensure all file paths, variable names, identifiers, and command options are exactly as provided
   - Do not output complete Stata code; only define the function as per the requirements
   - Maintain the exact specified display format for all output results
test_cases: local checks N1 17540 r2_1 0.6782 beta_DID_1 0.7099 se_DID_1 0.1683
local tolerance = 0.0001

forvalues i = 1(2)7 { 
    local name: word `i' of `checks'
    local expected: word `=`i'+1' of `checks'
    
    local diff = abs(``name'' - `expected')
    
    if `diff' > `tolerance' {
        display as error "Wrong"
        exit 1
    }
    else {
        display "Completely right!"
    }
}


step_number: 2
step_description_prompt: Please write Stata code to create a function named did_analysis

Function Definition:
This function performs a sequence of difference-in-differences (DID) regression analyses using the `reghdfe` command in Stata. It estimates three separate regression models with specified fixed effects and clustered standard errors, extracts key model statistics (number of observations, R-squared, coefficient, standard error, t-statistic, p-value for the `l.DID` variable, and coefficients for control variables if provided), and displays these results in a standardized format. The function accepts a local macro `x` that contains a list of control variables as input.

Algorithm Flow:
1. First Regression: Estimate a model where the dependent variable is `IN`, independent variables are `l.DID` plus all control variables in the local macro `x`. Absorb fixed effects `c`, `y`, and `iy`. Use clustered standard errors at the `provy` level.
2. Extract post-estimation statistics from the first regression:
   - Number of observations (`N2` = e(N))
   - Model R-squared (`r2_2` = e(r2))
   - Coefficient of `l.DID` (`beta_DID_2` = _b[l.DID])
   - Standard error of `l.DID` (`se_DID_2` = _se[l.DID])
   - Calculate t-statistic for `l.DID` as `t_DID_2 = beta_DID_2 / se_DID_2`
   - Calculate two-tailed p-value as `p_DID_2 = 2 * ttail(e(df_r), abs(t_DID_2))`
   - Extract coefficients for each control variable in `x` (only if the variable name is non-empty) and store them in local macros named `beta_[var]_2`
3. Display results for the first regression in the following order:
   - "IN ~ l.DID"
   - The value of `N2`
   - "R²: `r2_2`"
   - The coefficient of `l.DID` followed by its standard error and p-value in the format: "`beta_DID_2` (se=`se_DID_2', p=`p_DID_2')"
4. Second Regression: Estimate a model where the dependent variable is `IA`, independent variable is `l.DID` only (no control variables). Use the same fixed effects (`c`, `y`, `iy`) and clustered standard errors (`provy`) as the first regression.
5. Extract post-estimation statistics from the second regression:
   - `N3` = e(N), `r2_3` = e(r2), `beta_DID_3` = _b[l.DID], `se_DID_3` = _se[l.DID]
   - Calculate `t_DID_3 = beta_DID_3 / se_DID_3` and `p_DID_3 = 2 * ttail(e(df_r), abs(t_DID_3))`
6. Display results for the second regression:
   - "IA ~ l.DID"
   - `N3`
   - `r2_3`
   - "`beta_DID_3` (se=`se_DID_3', p=`p_DID_3')"
7. Third Regression: Estimate a model where the dependent variable is `IA`, independent variables are `l.DID` plus control variables in `x`. Use the same fixed effects and clustered standard errors as before.
8. Extract post-estimation statistics from the third regression:
   - `N4` = e(N), `r2_4` = e(r2), `beta_DID_4` = _b[l.DID], `se_DID_4` = _se[l.DID]
   - Calculate `t_DID_4 = beta_DID_4 / se_DID_4` and `p_DID_4 = 2 * ttail(e(df_r), abs(t_DID_4))`
   - Extract coefficients for each control variable in `x` (if non-empty) into local macros `beta_[var]_4`
9. Display results for the third regression:
   - "IA ~ l.DID"
   - `N4`
   - "R²: `r2_4`"
   - "`beta_DID_4` (se=`se_DID_4', p=`p_DID_4')"

Mathematical Principles:
- All regressions are linear fixed effects models estimated using the `reghdfe` command, which removes fixed effects (`c`, `y`, `iy`) by demeaning the variables, eliminating the need to estimate these fixed effects directly.
- Clustered standard errors (`vce(cluster provy)`) adjust for heteroskedasticity and within-group correlation at the `provy` level, providing more robust inference.
- The t-statistic for the `l.DID` coefficient is the ratio of the coefficient estimate to its standard error, testing the null hypothesis that the coefficient is zero.
- The two-tailed p-value is computed using the t-distribution with residual degrees of freedom (`e(df_r)`), where `ttail(df, t)` gives the upper tail probability, multiplied by 2 to account for both tails of the distribution.
- R-squared (`e(r2)`): Measures the proportion of variance in the dependent variable that is explained by the independent variables and fixed effects included in the model.

Technical Requirements:
- Use only standard Stata syntax; do not use `program define` to create a formal program.
- Utilize the `reghdfe` command for all three regression models with the specified `absorb` and `vce` options.
- Use Stata's built-in post-estimation macros (e(N), e(r2), e(df_r)) to extract model-level statistics.
- Use `_b[varname]` and `_se[varname]` to extract coefficient estimates and their standard errors.
- Use local macros to store all extracted statistics for later display.
- Include a check for non-empty variable names when extracting coefficients for control variables in `x`.

Notes:
- Ensure that the local macro `x` is defined before executing the code, containing the list of control variables (if any; it can be empty).
- The variables `IN`, `IA`, `l.DID`, `c`, `y`, `iy`, and `provy` must exist in the active dataset.
- The display output must match the specified format exactly, including labels for each regression and the "R²: " label for the first and third regressions.
- Do not add any additional functionality or output that is not specified in the algorithm flow.
test_cases: local checks N2 17369 r2_2 0.6825 beta_DID_2 0.5186 se_DID_2 0.1923 N3 17540 r2_3 0.6008 beta_DID_3 300.7893 se_DID_3 103.8747 N4 17369 r2_4 0.6056 beta_DID_4 342.8544 se_DID_4 183.8046
local tolerance = 0.0001

forvalues i = 1(2)23 { 
    local name: word `i' of `checks'
    local expected: word `=`i'+1' of `checks'
    
    local diff = abs(``name'' - `expected')
    
    if `diff' > `tolerance' {
        display as error "Wrong"
        exit 1
    }
    else {
        display "Completely right!"
    }
}


step_number: 3
step_description_prompt: Write Stata code to implement an analysis module named did_impact_assessment that performs the following steps without using `program define` syntax, adhering strictly to standard Stata command structures:

### 1. Function Definition
This module accepts a pre-defined macro variable $x (a list of control variables, may be empty) as input, executes three distinct fixed-effects regression analyses using the `reghdfe` command, extracts key regression statistics, and displays the results in a specified format.

### 2. Algorithm Flow
#### a. First Regression Analysis
- **Dependent Variable**: RDF
- **Independent Variables**: l.DID, all variables in $x
- **Fixed Effects to Absorb**: c, y, iy
- **Standard Error Specification**: Cluster-robust standard errors clustered by `provy`
- **Extract & Calculate Metrics**:
  - Sample size (store as local macro `N5`), R-squared (store as local macro `r2_5`)
  - Coefficient, standard error for l.DID (store as `beta_DID_5`, `se_DID_5`)
  - T-statistic: `t_DID_5 = beta_DID_5 / se_DID_5`; two-tailed p-value: `p_DID_5 = 2*ttail(e(df_r), abs(t_DID_5))`
  - For each non-empty variable in $x, store its coefficient as a local macro named `beta_`var'_5`
- **Display Exact Output**:
  ```
  "RDF ~ l.DID"
  [Value of N5]
  "R²: [Value of r2_5]"
  "[Value of beta_DID_5] (se=[Value of se_DID_5], p=[Value of p_DID_5])"
  ```

#### b. Second Regression Analysis
- **Dependent Variable**: RDP
- **Independent Variables**: l.DID, all variables in $x
- **Fixed Effects & Standard Error**: Same as the first regression
- **Extract & Calculate Metrics**:
  - Sample size (`N6`), R-squared (`r2_6`)
  - Coefficient, standard error for l.DID (`beta_DID_6`, `se_DID_6`), along with its t-statistic (`t_DID_6`) and two-tailed p-value (`p_DID_6`)
  - For each non-empty variable in $x, store its coefficient as `beta_`var'_6`
- **Display Exact Output**:
  ```
  "RDP ~ l.DID"
  [Value of N6]
  "R²: [Value of r2_6]"
  "[Value of beta_DID_6] (se=[Value of se_DID_6], p=[Value of p_DID_6])"
  ```

#### c. Third Regression Analysis
- **Dependent Variable**: IN
- **Independent Variables**: DID_lag1, interaction term `c.DID_lag1#c.KDM`, KDM, all variables in $x
- **Fixed Effects & Standard Error**: Same as the first regression
- **Extract & Calculate Metrics**:
  - Sample size (`N7`), R-squared (`r2_7`)
  - Coefficient, standard error for DID_lag1 (`beta_DIDlag1_7`, `se_DIDlag1_7`), interaction term (`beta_interact_7`, `se_interact_7`), KDM (`beta_KDM_7`, `se_KDM_7`)
  - T-statistics and two-tailed p-values for each of the three key variables using the formula: `t_var = coeff / se; p_var = 2*ttail(e(df_r), abs(t_var))`
  - For each non-empty variable in $x, store its coefficient as `beta_`var'_7`
- **Display Exact Output**:
  ```
  "IN ~ DID_lag1 + DID_lag1#KDM + KDM"
  [Value of N7]
  "R²: [Value of r2_7]"
  "[Value of beta_DIDlag1_7] (se=[Value of se_DIDlag1_7], p=[Value of p_DIDlag1_7])"
  "[Value of beta_interact_7] (se=[Value of se_interact_7], p=[Value of p_interact_7])"
  "[Value of beta_KDM_7] (se=[Value of se_KDM_7], p=[Value of p_KDM_7])"
  ```

### 3. Mathematical Principles
- All regressions use linear regression with high-dimensional fixed effects via `reghdfe` to control for unobserved heterogeneity in c, y, and iy.
- Cluster-robust standard errors adjust the variance-covariance matrix to account for within-group correlation across observations in the same `provy` cluster.
- Two-tailed p-values are computed using the t-distribution with residual degrees of freedom (`e(df_r)`) from each regression.

### 4. Technical Requirements
- Use only standard Stata commands and local macros to store/retrieve regression results; do not use `program define` syntax.
- Strictly replicate the exact regression specifications (dependent variables, independent variables, fixed effects, standard error clustering) for each step.
- Ensure all extracted statistics and calculated values match the variables and formulas specified.
- Display output exactly as formatted in the algorithm flow, with no additional text or modifications.
- The macro variable $x must be predefined in the Stata environment before executing this module, representing a list of control variables (can be empty).

### 5. Notes
- Ensure all variables referenced (RDF, RDP, IN, l.DID, DID_lag1, KDM, c, y, iy, provy) exist in the active Stata dataset.
- Do not add any extra regressions, variables, or statistical metrics not specified in the requirements.
- Do not alter the order of regression execution or result display.
test_cases: local checks N5 15279 r2_5 0.8563 beta_DID_5 0.4552 se_DID_5 0.1326 N6 12724 r2_6 0.9227 beta_DID_6 72.1042 se_DID_6 20.3493 N7 12425 r2_7 0.7016 beta_DIDlag1_7 1.3143 se_DIDlag1_7 0.2948 beta_interact_7 -3.4423 se_interact_7 0.4459 beta_KDM_7 0.9162 se_KDM_7 0.2991
local tolerance = 0.0001

forvalues i = 1(2)23 { 
    local name: word `i' of `checks'
    local expected: word `=`i'+1' of `checks'
    
    local diff = abs(``name'' - `expected')
    
    if `diff' > `tolerance' {
        display as error "Wrong"
        exit 1
    }
    else {
        display "Completely right!"
    }
}

step_number: 4
step_description_prompt: Please write Stata code to create a code block named reg_ia_interaction_report (using standard Stata syntax without `program define') that performs the following:

### Function Definition
This code block executes a fixed-effects regression with an interaction term, extracts key statistical results, calculates t-statistics and two-tailed p-values, and displays the results in a standardized formatted output.

### Algorithm Flow
1. Execute a fixed-effects regression using the `reghdfe` command with the following specifications:
   - Dependent variable: `IA`
   - Independent variables: `DID_lag1`, continuous interaction term `c.DID_lag1#c.KDM`, `KDM`, plus control variables stored in the macro `$x`
   - Absorb fixed effects: `c`, `y`, `iy`
   - Use cluster-robust standard errors clustered by the variable `provy`
2. Store the following regression outputs in local macros:
   - Number of observations: `e(N)` as `N8`
   - R-squared: `e(r2)` as `r2_8`
   - Coefficient of `DID_lag1`: `_b[DID_lag1]` as `beta_DIDlag1_8`; its standard error: `_se[DID_lag1]` as `se_DIDlag1_8`
   - Coefficient of `c.DID_lag1#c.KDM`: `_b[c.DID_lag1#c.KDM]` as `beta_interact_8`; its standard error: `_se[c.DID_lag1#c.KDM]` as `se_interact_8`
   - Coefficient of `KDM`: `_b[KDM]` as `beta_KDM_8`; its standard error: `_se[KDM]` as `se_KDM_8`
3. Calculate t-statistics for each coefficient as the ratio of the coefficient to its standard error, store in local macros:
   - `t_DIDlag1_8` = `_b[DID_lag1]/_se[DID_lag1]`
   - `t_interact_8` = `_b[c.DID_lag1#c.KDM]/_se[c.DID_lag1#c.KDM]`
   - `t_KDM_8` = `_b[KDM]/_se[KDM]`
4. Calculate two-tailed p-values using the `ttail()` function with residual degrees of freedom `e(df_r)`:
   - `p_DIDlag1_8` = `2*ttail(e(df_r), abs(`t_DIDlag1_8'))`
   - `p_interact_8` = `2*ttail(e(df_r), abs(`t_interact_8'))`
   - `p_KDM_8` = `2*ttail(e(df_r), abs(`t_KDM_8'))`
5. Display results in the exact following order and format:
   - First line: "IA ~ DID_lag1 + DID_lag1#KDM + KDM"
   - Second line: The value of `N8`
   - Third line: "R²: `r2_8'"
   - Fourth line: "`beta_DIDlag1_8' (se=`se_DIDlag1_8', p=`p_DIDlag1_8')"
   - Fifth line: "`beta_interact_8' (se=`se_interact_8', p=`p_interact_8')"
   - Sixth line: "`beta_KDM_8' (se=`se_KDM_8', p=`p_KDM_8')"

### Mathematical Principles
- The regression model is specified as:
  \( IA = \beta_0 + \beta_1 \times DID\_lag1 + \beta_2 \times DID\_lag1 \times KDM + \beta_3 \times KDM + \Gamma'X + \alpha_c + \alpha_y + \alpha_{iy} + \epsilon \)
  where \( X \) is the set of control variables in `$x`, \( \alpha_c, \alpha_y, \alpha_{iy} \) are fixed effects for `c`, `y`, `iy` respectively, and \( \epsilon \) is the error term.
- Cluster-robust standard errors adjust for within-cluster correlation among observations grouped by `provy`.
- T-statistic for each coefficient is \( t = \hat{\beta} / SE(\hat{\beta}) \), where \( SE(\hat{\beta}) \) is the cluster-robust standard error.
- Two-tailed p-value is \( 2 \times P(T > |t|) \), where \( T \) follows a t-distribution with residual degrees of freedom `e(df_r)`.

### Technical Requirements
- Must use the `reghdfe` command; ensure the `reghdfe` package is installed in Stata prior to execution.
- The macro `$x` must be pre-defined with the list of control variables to include in the regression.
- Specify standard errors with `vce(cluster provy)` to enable cluster-robust inference.
- Absorb fixed effects using the `absorb(c y iy)` option in `reghdfe`.
- Use Stata's built-in `_b[]` and `_se[]` to extract coefficients and standard errors, and `e()` macros for metadata (N, R², df_r).
- Calculate p-values exactly as specified using the `ttail()` function.
- Follow the exact format and order for all `display` commands; do not alter the output structure.

### Notes
- Do not use `program define` syntax; implement the code as a sequential set of standard Stata commands.
- All local macros must be properly defined before they are used in calculations or display commands.
- Ensure the continuous interaction term is specified as `c.DID_lag1#c.KDM` to indicate both variables are continuous.
- Maintain strict adherence to the original code's logic; do not add or remove any functionality.
test_cases: local checks N8 12425 r2_8 0.6169 beta_DIDlag1_8 698.8943 se_DIDlag1_8 287.4816 beta_interact_8 -1032.0410 se_interact_8 280.6054 beta_KDM_8 605.4290 se_KDM_8 227.5052
local tolerance = 0.0001

forvalues i = 1(2)15 { 
    local name: word `i' of `checks'
    local expected: word `=`i'+1' of `checks'
    
    local diff = abs(``name'' - `expected')
    
    if `diff' > `tolerance' {
        display as error "Wrong"
        exit 1
    }
    else {
        display "Completely right!"
    }
}

General solution:

General tests: