Session 1: Surveys, IO and SAM
If you are reading this, you are probably interested in the Microeconometrics path of Research methods II.
This course assumes you are familiar with the basics of econometrics and statistics.
This course will focus on tools that are commonly used in empirical research in economics:
We will emphasize the use of Household Surveys (with the issues it entails)
And focus on applied econometrics using cross-sectional data, for distributional analysis and policy simulation.
Thus, we will do much more use of empirical tools and software than we did in Research methods I.
We will have 7 Sessions, and 2 homeworks .
A Good survey is one that allows you to obtain estimates of statistics of interest for the population with “Tolerable” levels of Accuracy.
To do this, you need to have a good sampling design (representation and “independence of the population”) and a good questionnaire design (questions that are clear and easy to answer).
A good survey needs to be representative of the population of interest. To do this appropriately, data will be collected based on a frame that will be used to select the sample.
Each observation in the “frame” has the same probability of being selected in the sample.
It may be difficult to implement in practice, because of cost and logistics. (distance)
It could also have problems of representativeness for small groups. (rare events)
Once you have your data, you can start analyzing it by simply applying your survey weights.
However, when considering the estimation of precision of the estimates (Variance and Standard Errors), there are two approaches that are important to consider:
Finite Population Approach:
Superpopulation Approach:
For practical purposes, the difference between the two approaches is not that important.
Where \(fpc = \frac{N-n}{N-1}\) is the finite population correction.
The previous formulas did not account for weights.
Weights are factors used to “reweight/expand” the sample to make it representative of the population.
They can are typically related to the inverse of the probability of selection.
Simply said, a weight \(w_i=\frac{1}{n*\pi_i}\) is a measure of how many observations in the population are represented by observation \(i\) in the sample.
But how do we account for weights in the formulas?
Population: \[\hat N = \sum_{i=1}^n w_i\]
Normalized weights \[v_i = \frac{n w_i}{ \sum w_i} \rightarrow E(v_i) = 1\]
Mean: \[\hat{\mu}_y=\bar{y} = \frac{1}{\hat N}\sum_{i=1}^n w_i y_i = \frac{1}{n}\sum v_i y_i\]
Variances are a bit more complicated. Normally you would consider:
\[Var(\bar y) = \frac{1}{n} \hat \sigma^2_y =\frac{1}{n} \frac{1}{n-1} \sum_{i=1}^n v_i (y_i - \bar y)^2\]
However, with survey weights you need to consider something else: \[Var(\bar y) = \frac{1}{n} \sum_{i=1}^n v_i^2 (y_i - \bar y)^2\]
Which is similar to “robust” Standard errors in OLS.
What about Clusters and Strata?
Lets use data two Examples.
First Labor Force Survey: Oaxaca
Second National Health and Nutrition Examination Survey (NHANES)
frause oaxaca, clear
** Create Weights <- This will be provided
sum wt, meanonly
replace wt = round(wt/r(min))
gen wage = exp(lnwage)
tab wt(Excerpt from the Swiss Labor Market Survey 1998)
(1,647 real changes made)
(213 missing values generated)
sampling |
weights | Freq. Percent Cum.
------------+-----------------------------------
1 | 489 29.69 29.69
2 | 924 56.10 85.79
3 | 160 9.71 95.51
4 | 64 3.89 99.39
5 | 8 0.49 99.88
6 | 2 0.12 100.00
------------+-----------------------------------
Total | 1,647 100.00Weights Distribution
sampling |
weights | Freq. Percent Cum.
------------+-----------------------------------
1 | 489 29.69 29.69
2 | 924 56.10 85.79
3 | 160 9.71 95.51
4 | 64 3.89 99.39
5 | 8 0.49 99.88
6 | 2 0.12 100.00
------------+-----------------------------------
Total | 1,647 100.00Summary Statistics:
wage
-------------------------------------------------------------
Percentiles Smallest
1% 3.907204 1.661434
5% 11.8007 1.873127
10% 17.4216 2.197802 Obs 1,434
25% 23.19902 2.442003 Sum of wgt. 1,434
50% 30.08896 Mean 32.39167
Largest Std. dev. 16.12498
75% 38.5662 137.3627
90% 49.95005 152.6252 Variance 260.0151
95% 58.71271 164.8352 Skewness 2.486954
99% 85.47008 192.3077 Kurtosis 18.23702
wage
-------------------------------------------------------------
Percentiles Smallest
1% 3.453689 1.661434
5% 7.370678 1.873127
10% 15.83933 2.197802 Obs 1,434
25% 21.72247 2.442003 Sum of wgt. 2,686
50% 29.48271 Mean 31.5322
Largest Std. dev. 16.32811
75% 38.46154 137.3627
90% 49.95005 152.6252 Variance 266.6071
95% 58.11497 164.8352 Skewness 2.081806
99% 85.47008 192.3077 Kurtosis 14.68647Accounting for weights for summary Statistics
Mean estimation Number of obs = 1,434
--------------------------------------------------------------
| Mean Std. err. [95% conf. interval]
-------------+------------------------------------------------
wage | 32.39167 .4258186 31.55637 33.22696
--------------------------------------------------------------
Mean estimation Number of obs = 1,434
--------------------------------------------------------------
| Mean Std. err. [95% conf. interval]
-------------+------------------------------------------------
wage | 31.5322 .4765835 30.59733 32.46708
--------------------------------------------------------------
Mean estimation Number of obs = 1,434
---------------------------------------------------------------
| Mean Std. err. [95% conf. interval]
--------------+------------------------------------------------
c.wage@female |
0 | 33.87649 .6152059 32.66969 35.08329
1 | 28.76133 .722173 27.3447 30.17796
---------------------------------------------------------------Tables and cross tables
| sex of respondent
years of | (1=female)
education | 0 1 | Total
-----------+----------------------+----------
5 | 6 23 | 29
9 | 47 93 | 140
9.75 | 8 26 | 34
10 | 10 42 | 52
10.5 | 376 447 | 823
11.5 | 7 14 | 21
12 | 111 87 | 198
12.5 | 56 80 | 136
15 | 60 13 | 73
17.5 | 78 63 | 141
-----------+----------------------+----------
Total | 759 888 | 1,647
| sex of respondent
years of | (1=female)
education | 0 1 | Total
-----------+----------------------+----------
5 | 17 45 | 62
9 | 104 181 | 285
9.75 | 14 52 | 66
10 | 22 80 | 102
10.5 | 725 833 | 1,558
11.5 | 19 27 | 46
12 | 201 148 | 349
12.5 | 108 157 | 265
15 | 111 21 | 132
17.5 | 149 111 | 260
-----------+----------------------+----------
Total | 1,470 1,655 | 3,125 Better approach: svyset
webuse nhanes2f, clear
svyset psuid /// Cluster
[pweight=finalwgt], /// Survey Weight as Inverse of Prob of Selection
strata(stratid) // Strata Identifier
mean zinc
mean zinc [pw=finalwgt]
svy: mean zinc
Sampling weights: finalwgt
VCE: linearized
Single unit: missing
Strata 1: stratid
Sampling unit 1: psuid
FPC 1: <zero>
Mean estimation Number of obs = 9,189
--------------------------------------------------------------
| Mean Std. err. [95% conf. interval]
-------------+------------------------------------------------
zinc | 86.51518 .1510744 86.21904 86.81132
--------------------------------------------------------------
Mean estimation Number of obs = 9,189
--------------------------------------------------------------
| Mean Std. err. [95% conf. interval]
-------------+------------------------------------------------
zinc | 87.18207 .1828747 86.82359 87.54054
--------------------------------------------------------------
(running mean on estimation sample)
Survey: Mean estimation
Number of strata = 31 Number of obs = 9,189
Number of PSUs = 62 Population size = 104,176,071
Design df = 31
--------------------------------------------------------------
| Linearized
| Mean std. err. [95% conf. interval]
-------------+------------------------------------------------
zinc | 87.18207 .4944827 86.17356 88.19057
--------------------------------------------------------------Consider two groups with the following characteristics:
| Group | N | mean | Var |
|---|---|---|---|
| 1 | 100 | 45 | 32.56 |
| 2 | 150 | 55 | 21.97 |
Is the difference in means statistically significant?
Test \(H_0: \mu_1 = \mu_2\) vs \(H_1: \mu_1 \neq \mu_2\)
\[t = \frac{\mu_2 - \mu_1}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} =\frac{10}{\sqrt{\frac{32.56}{100}+\frac{21.97}{150}}}=\frac{10}{\sqrt{.47207}} = 14.55453 \]
If \(t\) is large enough, we can reject the null hypothesis.
But this does not work if you have weights…
frause oaxaca, clear
** Create Weights <- This will be provided
sum wt, meanonly
replace wt = round(wt/r(min))
gen wage = exp(lnwage)
reg wage i.female [pw=wt],(Excerpt from the Swiss Labor Market Survey 1998)
(1,647 real changes made)
(213 missing values generated)
(sum of wgt is 2,686)
Linear regression Number of obs = 1,434
F(1, 1432) = 29.05
Prob > F = 0.0000
R-squared = 0.0244
Root MSE = 16.133
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
1.female | -5.115156 .9490209 -5.39 0.000 -6.976776 -3.253536
_cons | 33.87649 .6154206 55.05 0.000 32.66927 35.08371
------------------------------------------------------------------------------svy: regress for complex designs.see here for additional examples
IO tables stand for Input-Output tables. They are a way to represent the production structure of an economy.
They provide a Static representation of the Economy
Each Row represents the production of a sector, and each column represents the use of that production by other sectors as Inputs.
The information it contains represent a snapshot of the economy at a given point in time.
It can be used to simulate changes in production, labor demand, and total production in the economy, under specific assumptions (production function)
Consider a Economy with K=3 sectors: Agriculture, Manufacturing and Services, with a Final consumer agent (households)
Each sector (i) produces a good (\(X_i\)), which is sold to other sectors or the final consumer.
\(X_{ij}\) is the quantity of goods sector \(i\) sells to sector \(j\), and \(y_i\) the final consumption by households.
\(X_{ji}\) is also the quantity of goods sector \(i\) uses from sector \(j\) as inputs.
\(L_i\) is the amount of labor used by sector \(i\).
In a simple Economy, (value) Total labor demand is equal to total Household income. \(L_a + L_m + L_s = L = y_a + y_m + y_s\)
This simple Economy can be represented as follows:
\[ \begin{aligned} X_{11} + X_{12} + X_{13} + Y_{1} &= X_{1} \\ X_{21} + X_{22} + X_{23} + Y_{2} &= X_{2} \\ X_{31} + X_{32} + X_{33} + Y_{3} &= X_{3} \\ L_1 + L_2 + L_3 + 0 &= L \end{aligned} \]
| S1 | S2 | S3 | HH | ||
|---|---|---|---|---|---|
| Supply: | |||||
| S1 | \(X_{11}\) | \(X_{12}\) | \(X_{13}\) | \(Y_{1}\) | \(X_{1}\) |
| S2 | \(X_{21}\) | \(X_{22}\) | \(X_{23}\) | \(Y_{2}\) | \(X_{2}\) |
| S3 | \(X_{31}\) | \(X_{32}\) | \(X_{33}\) | \(Y_{3}\) | \(X_{3}\) |
| HH | \(L_1\) | \(L_2\) | \(L_3\) | 0 | \(L\) |
From the consumption/inputs Side, we could also write the equations as:
\[ \begin{aligned} X_{11} &+ X_{21} &+ X_{31} &+ L_{1} &= X_{1} \\ X_{12} &+ X_{22} &+ X_{32} &+ L_{2} &= X_{2} \\ X_{13} &+ X_{23} &+ X_{33} &+ L_{3} &= X_{3} \\ Y_1 &+ Y_2 &+ Y_3 & &= Y \end{aligned} \]
And from here we can get the technical coefficients:
\[ \begin{aligned} a_{11} X_1 &+ a_{21} X_1 &+ a_{31} X_1 &+ \lambda_{1} X_1 &= X_1 \\ a_{12} X_2 &+ a_{22} X_2 &+ a_{32} X_2 &+ \lambda_{2} X_2 &= X_2 \\ a_{13} X_3 &+ a_{23} X_3 &+ a_{33} X_3 &+ \lambda_{3} X_3 &= X_3 \\ \delta_1 Y &+ \delta_2 Y & + \delta_3 Y & &= Y \end{aligned} \]
Where \(a_{ij} = \frac{X_{ij}}{X_j}\) and \(\lambda_i = \frac{L_i}{X_i}\)
\[a_{1i}+a_{2i}+a_{3i}+\lambda_i = 1 \ \& \ \delta_1+\delta_2+\delta_3 =1\]
With this, we can write the IO table
\[ \begin{aligned} a_{11} X_1 &+ a_{12} X_2 &+ a_{13} X_3 &+ \delta_1 Y &= X_{1} \\ a_{21} X_1 &+ a_{22} X_2 &+ a_{23} X_3 &+ \delta_2 Y &= X_{2} \\ a_{31} X_1 &+ a_{32} X_2 &+ a_{33} X_3 &+ \delta_3 Y &= X_{3} \\ \lambda_1 X_1 &+ \lambda_2 X_2 &+ \lambda_3 X_3 &\ \ &= L \end{aligned} \]
Or into Matrix Form
\[ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \lambda_1 & \lambda_2 & \lambda_3 \end{pmatrix} \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} + \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \\ 0 \end{pmatrix}= \begin{pmatrix} X_1 \\ X_2 \\ X_3 \\ L \end{pmatrix} \]
Solve for Production sectors:
\[ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} + \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix}= \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} \]
\[ \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix}=I \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} - A \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} = (I-A) \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} \]
Finally:
\[ \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} = (I-A)^{-1} \begin{pmatrix} Y_1 \\ Y_2 \\ Y_3 \end{pmatrix} \rightarrow \begin{pmatrix} \Delta X_1 \\ \Delta X_2 \\ \Delta X_3 \end{pmatrix} = (I-A)^{-1} \begin{pmatrix} \Delta Y_1 \\ \Delta Y_2 \\ \Delta Y_3 \end{pmatrix} \]
| Sector | Agriculture | Manufacturing | Services | Final Consumer |
|---|---|---|---|---|
| Agriculture | 102 | 103 | 153 | 129 |
| Manufacturing | 133 | 124 | 77 | 99 |
| Services | 71 | 92 | 51 | 165 |
| Households | 181 | 114 | 98 | 0 |
S1: What is the total production of each sector?
S2: What are the technical coefficients?
| Sector | Agriculture | Manufacturing | Services | Final Consumer |
|---|---|---|---|---|
| Agriculture | \(a_{11}\) = 0.209 | 0.238 | 0.404 | 0.328 |
| Manufacturing | 0.273 | 0.286 | 0.203 | 0.252 |
| Services | 0.146 | 0.212 | 0.135 | 0.420 |
| Households | 0.372 | 0.263 | 0.259 | 0.000 |
This captures a snapshot of an economy. And could use to simulate changes in production and labor demand.
S3. How much would production change if the final consumer demand for Agriculture increases in 20%?
\[\Delta X = (I-A)^-1 \begin{pmatrix} 0.2 * 129 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 45.54 \\ 21.08 \\ 12.84 \end{pmatrix} \]
S4: How much would labor demand change if the final consumer demand for Agriculture increases in 20%? \[\Delta L_i = \lambda_i * \Delta X_i\]
\(\Delta L_1 = 0.372 * 45.54 = 16.95 ; \Delta L_2 = 0.263 * 21.08 = 5.544 ; \Delta L_3 = 0.259 * 12.84 = 3.326\)
Statamata: Input Data
Estimate Total Production
Estimate Technical Coefficients
1 2 3
+-------------------------------------------+
1 | .2094455852 .2378752887 .4036939314 |
2 | .273100616 .2863741339 .2031662269 |
3 | .1457905544 .2124711316 .1345646438 |
+-------------------------------------------+Estimate Change in Demand: 20% increase in Agriculture
1
+--------+
1 | 25.8 |
2 | 0 |
3 | 0 |
+--------+Estimate Change in Production
1
+---------------+
1 | 45.54130481 |
2 | 21.08637814 |
3 | 12.84872246 |
+---------------+Estimate Change in Labor Demand
SAM can be thought as an upgraded version of IO-tables.
They are a way to organize information about the production structure of an economy, but also the distribution of resources.
You can also use it as basis for a plausible model of the economy.
| Production | Consumption | Accumulation | Totals | |
|---|---|---|---|---|
| Production | \(C\) | \(I\) | \(C+I\) | |
| Consumption | \(Y\) | \(Y\) | ||
| Accumulation | \(S\) | \(S\) | ||
| Totals | \(Y\) | \(C+S\) | \(I\) |
| S1 | S2 | S3 | S4 | S5 | Totals | |
|---|---|---|---|---|---|---|
| S1: Prod | \(C\) | \(G\) | \(I\) | \(E\) | \(C+G+I+E\) | |
| S2: HH | \(Y\) | \(Y\) | ||||
| S3: Gov | \(Tx\) | \(Tx\) | ||||
| S4: K acc | \(S_h\) | \(S_g\) | \(S_f\) | \(S_h+S_g+S_f\) | ||
| S5: RofW | \(M\) | \(M\) | ||||
| Totals | \(Y+M\) | \(C+S_h+Tx\) | \(G+S_g\) | \(I\) | \(E+S_f\) |
Here \(S1\) and \(S2\) is what we had in the IO table. Thus, we could further expand the SAM to include more sectors and agents.
| S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 | |
|---|---|---|---|---|---|---|---|---|
| S1: Act | \(Gds\) | |||||||
| S2: Commod | \(IntGds\) | \(C\) | \(G\) | \(I\) | \(E\) | |||
| S3: Factors | \(VA\) | \(FE\) | ||||||
| S4: Enter | \(Prof\) | \(ETr\) | ||||||
| S5: HH | \(Wage\) | \(DProf\) | \(Tr\) | \(REM\) | ||||
| S6: Gov | \(ITx\) | \(FTx\) | \(ETx\) | \(DTx\) | \(Tarf\) | |||
| S7: K acc | \(RetY\) | \(S_h\) | \(S_g\) | \(KTrM\) | ||||
| S8: RofW | \(M\) | \(FM\) | \(REMA\) | \(TrA\) | \(KtrA\) |