# Compute Jackknife Coefficient Estimates in SAS¶

In certain scenarios, we want to estimate a model's parameters on the sample for each observation with itself excluded. This can be achieved by estimating the model repeatedly on the leave-one-out samples but is very inefficient. If we estimate the model on the full sample, however, the coefficient estimates will certainly be biased. Thankfully, we have the Jackknife method to correct for the bias, which produces the Jackknifed coefficient estimates for each observation.

## Variable Definition¶

Variable Definition
$b(i)$ the parameter estimates after deleting the $i$th observation
$s^2(i)$ the variance estimate after deleting the $i$th observation
$X(i)$ the $X$ matrix without the $i$th observation
$\hat{y}(i)$ the $i$th value predicted without using the $i$th observation
$r_i = y_i - \hat{y}_i$ the $i$th residual
$h_i = x_i(X'X)^{-1}x_i'$ the $i$th diagonal of the projection matrix for the predictor space, also called the hat matrix
$RStudent =\frac{r_i}{s(i) \sqrt{1-h_i}}$ studentized residual
$(X'X)_{jj}$ the $(j,j)$th element of $(X'X)^{-1}$
$DFBeta_j = \frac{b_{j} - b_{(i)j}}{s(i)\sqrt{(X'X)_{jj}}}$ the scaled measures of the change in the $j$th parameter estimate calculated by deleting the $i$th observation

## Objective¶

Compute the coefficient estiamtes with the $i$th observation excluded from the sample, i.e. $b(i)$, or the Jackknifed coefficient estimate.

## Formula¶

From the table above, we can get that the $j$th Jackknifed coefficient estimate $b_{(i)j}$ without using the $i$th observation is:

b_{(i)j} = b_j - DFBeta_j \times s(i) \sqrt{(X'X)_{jj}}

Hence,

b_{(i)j} = b_j - DFBeta_j \times \frac{r_i}{RStudent\times \sqrt{1-h_i}} \sqrt{(X'X)_{jj}}

The good thing is that PROC REG produces the coefficient estimate $b_j$ for $j=1,2,...K$, where $K$ is the number of coefficients, and the INFLUENCE and I options produce the remaining statistics just enough to compute $b(i)$:

Variable Option in PROC REG or MODEL statement Name in the output dataset
$b_j$ Outest= option in PROC REG <jthVariable>
$r_i$ OutputStatistics= from INFLUENCE option in MODEL statement Residual
$RStudent$ OutputStatistics= from INFLUENCE option in MODEL statement RStudent
$h_i$ OutputStatistics= from INFLUENCE option in MODEL statement HatDiagnol
$DFBeta_j$ OutputStatistics= from INFLUENCE option in MODEL statement DFB_<jthVariable>
$(X'X)_{jj}$ InvXPX= from I option in MODEL statement <jthVariable>

## Example¶

### Discretionary accruals¶

Suppose we want to calculate the firm-level discretionary accruals for each year using the Jones (1991) model and Kothari et al (2005) model. For a firm $i$, we need to first estimate the model for the industry-year excluding firm $i$, then use the coefficient estimates to generate predicted accruals for firm $i$. The firm's discretionary accruals is the actual accruals minus the predicted accruals.

Below is an example PROC REG that produces three datasets named work.params, work.outstats and work.xpxinv, which contain sufficient statistics to compute the Jackknifed estimates and thus the predicted accruals.

  1 2 3 4 5 6 7 8 9 10 11 12 13 ods listing close; proc reg data=work.funda edf outest=work.params; /* industry-year regression */ by fyear sic2; /* id is necessary for later matching Jackknifed coefficients to firm-year */ id key; /* Jones Model */ Jones: model tac = inv_at_l drev ppe / noint influence i; /* Kothari Model with ROA */ Kothari: model tac = inv_at_l drevadj ppe roa / noint influence i; ods output OutputStatistics=work.outstats InvXPX=work.xpxinv; run; ods listing; 

Full SAS program for estimating 5 different measures of discretionary accruals:

SAS code for computing discretionary accruals
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 /* Use Jackknife method to compute discretionary accruals */ /* see https://mingze-gao.com/posts/compute-jackknife-coefficient-estimates-in-sas/ */ /* UseHribarCollinsTotalAccruals: - true: use Hribar-Collins Cashflow Total Accruals - false: use normal method */ %let UseHribarCollinsTotalAccruals = false; /* Include %array and %do_over */ filename do_over url "https://mingze-gao.com/utils/do_over.sas"; filename array url "https://mingze-gao.com/utils/array.sas"; %include do_over array; /* Winsorize macro */ filename winsor url "https://mingze-gao.com/utils/winsor.sas"; %include winsor; /* Earnings management models Author: Mingze (Adrian) Gao, Feb 2019 Modified based on the work by Joost Impink, March 2016 Models estimated (Note that the intercept a0 is removed in the modified code below): - Jones model, tac = a0 + a1 1/TAt-1 + a2chSales + a3PPE + a4ROA + error. - variable names DA_Jones - Modified Jones model, as Jones model, but using chSales - chREC to compute fitted values. - variable names DA_mJones - Kothari 2005, controlling for ROA, tac = a0 + a1 1/TAt-1 + a2(chSales - chREC) + a3PPE + a4ROA + error. - variable names DA_Kothari - Kothari 2005, performance matched, Jones model, difference in discretionary accruals between firm and closest firm in terms of (contemporaneous) roa - variable names DA_pmKothari_Jones - Kothari 2005, performance matched, modified Jones model, difference in discretionary accruals between firm and closest firm in terms of (contemporaneous) roa - variable names DA_pmKothari_mJones tac: Total accruals, computed as net profit after tax before extraordinary items less cash flows from operations 1/TAt-1: Inverse of beginning of year total assets chSales: Change in net sales revenue chREC: Change in net receivables PPE: Gross property, plant, and equipment ROA: Return on assets. Variables used Compustat Funda AT: Total assets IB: Income Before Extraordinary Items IBC: Income Before Extraordinary Items (Cash Flow) (used if IB is missing) OANCF: Operating Activities - Net Cash Flow PPEGT: Property, Plant and Equipment - Total (Gross) RECT: Receivables - Total SALE: Sales INVT: Inventories - Total LCO: Current Liabilities Other Total DP: Depreciation and Amortization ACO: Current Assets Other Total AP: Accounts Payable - Trade */ /* Get Funda variables */ %let fundaVars = at ib ibc oancf ppegt rect sale xidoc lco dp aco invt ap; data work.a_funda(keep=key gvkey fyear datadate sich &fundaVars); set comp.funda; if 1980 <= fyear <= 2018; /* Generic filter */ if indfmt='INDL' and datafmt='STD' and popsrc='D' and consol='C'; /* Firm-year identifier */ key = gvkey || fyear; /* Keep if sale > 0, at > 0 */ if sale > 0 and at > 0; /* Use Income Before Extraordinary Items (Cash Flow) if ib is missing */ if ib =. then ib=ibc; run; /* Lagged values for: at sale rect invt aco ap lco */ %let lagVars = at sale rect invt aco ap lco; /* Self join to get lagged values at_l, sale_l, rect_l */ proc sql; create table work.b_funda as select a.*, %do_over(values=&lagVars, between=comma, phrase=b.? as ?_l) from work.a_funda a, work.a_funda b where a.gvkey = b.gvkey and a.fyear-1 = b.fyear; quit; /* Construct additional variables */ data work.b_funda(compress=yes); set work.b_funda; /* 2-digit SIC */ sic2 = int(sich/100); /* variables */ if "&UseHribarCollinsTotalAccruals." eq "false" then tac = ((rect-rect_l)+(invt-invt_l)+(aco-aco_l)-(ap-ap_l)-(lco-lco_l)-dp)/at_l; /* Accruals ratio */ else tac = (ibc - oancf + xidoc)/at_l; /* Hribar Collins total cash flow accruals */ inv_at_l = 1 / at_l; drev = (sale - sale_l) / at_l; drevadj = (sale - sale_l)/at_l - (rect - rect_l)/at_l; ppe = ppegt / at_l; roa = ib / at_l; /* these variables may not be missing (cmiss counts missing variables)*/ *if cmiss (of tac inv_at_l drevadj ppe roa) eq 0; run; /* Optional winsorization before industry-year regression */ %let winsVars = tac inv_at_l drev drevadj ppe roa ; %winsor(dsetin=work.b_funda, dsetout=work.b_funda_wins, byvar=fyear, vars=&winsVars, type=winsor, pctl=1 99); /* Regression by industry-year edf(error degrees of freedom) + #params will equal the number of obs (no need for proc univariate to count) */ proc sort data=work.b_funda_wins; by fyear sic2; run; /* regressors */ %array(vars, values=inv_at_l drev ppe drevadj roa); ods listing close; proc reg data=work.b_funda_wins edf outest=work.c_parms; by fyear sic2; id key; /* Jones Model */ Jones: model tac = inv_at_l drev ppe / noint influence i; /* Kothari with ROA in model */ Kothari: model tac = inv_at_l drevadj ppe roa / noint influence i; ods output OutputStatistics=work.outstats InvXPX=work.xpxinv; run; ods listing; /* Compute discretionary accrual measures */ proc sql; /* Compute firm-year Jackknifed coefficient estimates */ create table work.xpxinv2 as /* Extract the diagnol elements of the symmetric inv(X'X) for each firm-year */ select fyear, sic2, model, %do_over(vars, phrase=sum(case when variable="?" then xpxinv else . end) as ?, between=comma) from (select fyear, sic2, model, variable, case %do_over(vars, phrase=when variable="?" then ?) else . end as xpxinv from work.xpxinv where variable ~= 'tac') group by fyear, sic2, model order by fyear, sic2, model; /* The difference between original coefficient estimates and the Jackknifed estimates */ create table work.bias as select a.fyear, a.sic2, a.model, a.key, %do_over(vars, phrase=a.DFB_?*(a.Residual/(a.RStudent*sqrt(1-a.HatDiagonal)))*sqrt(b.?) as bias_?, between=comma) from work.outstats as a left join work.xpxinv2 as b on a.fyear=b.fyear and a.sic2=b.sic2 and a.model=b.model order by a.fyear, a.sic2, a.model, a.key; /* Compute Jackknifed coefficient estimates by subtracting the bias from the original estimates */ create table work.Jackknifed_params as select a.fyear, a.sic2, a.model, a.key, %do_over(vars, phrase=b.? - a.bias_? as ?, between=comma), b._EDF_ from work.bias as a left join work.c_parms as b on a.fyear=b.fyear and a.sic2=b.sic2 and a.model=b._MODEL_ order by a.fyear, a.sic2, a.model, a.key; /* Compute discretionary accruals */ create table work.tmp as select distinct a.fyear, a.sic2, a.gvkey, a.key, /* Jones model at a minimum 8 obs (5 degrees of freedom + 3 params) */ sum(case when b.model eq 'Jones' and b._EDF_ ge 5 then a.tac - (%do_over(values=inv_at_l drev ppe, between=%str(+), phrase=a.? * b.?)) else . end) as DA_Jones, /* Modified Jones model: drev is used in first model, but drevadj is used to compute fitted value */ sum(case when b.model eq 'Jones' and b._EDF_ ge 5 then a.tac - (a.drevadj * b.drev + %do_over(values=inv_at_l ppe, between=%str(+), phrase=a.? * b.?)) else . end) as DA_mJones, /* Kothari model (with ROA in regression) at a minimum 8 obs (4 degrees of freedom + 4 params) */ sum(case when b.model eq 'Kothari' and b._EDF_ ge 4 then a.tac - (%do_over(values=inv_at_l drevadj ppe roa, between=%str(+), phrase=a.? * b.?)) else . end) as DA_Kothari from work.b_funda_wins as a left join work.Jackknifed_params as b on a.key=b.key group by a.key order by a.gvkey, a.fyear; /* Kothari performance matching: get DA_Jones (DA_mJones) accruals for the matched firm closest in ROA */ create table work.da_roa as select a.*, b.roa from work.tmp as a left join work.b_funda_wins as b on a.key=b.key; create table work.da_all as select a.*, /* gvkey of matched firm */ b.gvkey as gvkey_m, /* difference in ROA */ abs(a.roa - b.roa) as Difference, /* difference in DA_Jones */ a.DA_Jones - b.DA_Jones as DA_pmKothari_Jones, a.DA_mJones - b.DA_mJones as DA_pmKothari_mJones from work.da_roa as a left join work.da_roa as b on a.fyear = b.fyear and a.sic2 = b.sic2 /* same 2-digit SIC industry-year */ and a.key ne b.key /* not the same firm */ group by a.gvkey, a.fyear having Difference = min(Difference) /* keep best match for size difference */ order by gvkey, fyear; quit; /* drop possible multiple matches (with the same difference) in previous step */ proc sort data=work.da_all nodupkey; by key; run; %let DAVars = DA_Jones DA_mJones DA_Kothari DA_pmKothari_Jones DA_pmKothari_mJones; /* Winsorize discretionary accrual variables (Optional) */ %winsor(dsetin=work.da_all, dsetout=work.accruals_HribarCollins_&UseHribarCollinsTotalAccruals., byvar=fyear, vars=&DAVars, type=winsor, pctl=1 99); /* Means, medians for key variables */ proc means data=work.accruals_HribarCollins_&UseHribarCollinsTotalAccruals. n mean min median max; var &DAVars; run; 

Last update: June 10, 2020