Ingredients for statistical inference

1. Estimand \(\theta\)

2. Estimator \(\widehat \theta\)

Ingredients for statistical inference

1. Estimand \(\mu\)

2. Estimator \(\widehat \mu\)

Ingredients for statistical inference

1. Estimand \(\mu\)

2. Estimator \(\widehat \mu\)

3. Data \(X\)

Ingredients for statistical inference

1. Estimand \(\mu\)

2. Estimator \(\widehat \mu\)

3. Data \(X\)

4. Estimate \(\overline X\)

Ingredients for statistical inference

1. Estimand \(\mu\)

2. Estimator \(\widehat \mu\)

3. Data \(X\)

4. Estimate \(\overline X\)

5. Quantify uncertainty

Ingredients for statistical inference

1. Estimand \(\mu\)

2. Estimator \(\widehat \mu\)

3. Data \(X\)

4. Estimate \(\overline X\)

5. Quantify uncertainty

   • Confidence intervals (lab)
   • Hypothesis testing (week 8)

\(X \rightarrow \overline{X} \rightarrow \widehat{\mu} \xrightarrow{\text{hopefully!}} \mu\)

Why random sampling?

Independent, identically distributed

A collection of random variables is i.i.d. if they are:

1. Mutually independent

2. Identically distributed

Data

• Code
• Data

library(tidyverse)
# remotes::install_github("kjhealy/gssr")
library(gssr)

gss22 = gss_get_yr(2022)

# Question: 
# Do you remember for sure whether or not you voted in the last election?
gss = gss22 %>% 
  select(vote20) %>% 
  mutate(vote = ifelse(vote20 == 1, 1, 0)) %>% 
  drop_na()

     vote20 vote
1         1    1
2         1    1
3         1    1
4         1    1
5         1    1
6         1    1
7         2    0
8         1    1
9         1    1
10        1    1
11        1    1
12        1    1
13        1    1
14        1    1
15        1    1
16        1    1
17        1    1
18        1    1
19        2    0
20        1    1
21        3    0
22        1    1
23        1    1
24        1    1
25        1    1
26        1    1
27        1    1
28        1    1
29        1    1
30        1    1
31        1    1
32        1    1
33        1    1
34        1    1
35        2    0
36        1    1
37        1    1
38        1    1
39        1    1
40        2    0
41        1    1
42        3    0
43        1    1
44        2    0
45        1    1
46        1    1
47        1    1
48        1    1
49        2    0
50        1    1
51        1    1
52        3    0
53        2    0
54        1    1
55        1    1
56        2    0
57        1    1
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60        2    0
61        1    1
62        1    1
63        2    0
64        1    1
65        1    1
66        2    0
67        2    0
68        1    1
69        1    1
70        1    1
71        1    1
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73        1    1
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79        2    0
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3129      1    1
3130      1    1
3131      1    1
3132      1    1
3133      1    1
3134      1    1
3135      2    0
3136      1    1
3137      1    1
3138      1    1
3139      1    1
3140      1    1
3141      2    0
3142      1    1
3143      1    1
3144      1    1
3145      1    1
3146      1    1
3147      1    1
3148      1    1
3149      1    1
3150      2    0
3151      1    1
3152      1    1
3153      1    1
3154      1    1
3155      1    1
3156      1    1
3157      1    1
3158      1    1
3159      1    1
3160      1    1
3161      1    1
3162      2    0
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3164      1    1
3165      1    1
3166      1    1
3167      1    1
3168      1    1
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3170      1    1
3171      1    1
3172      1    1
3173      1    1
3174      1    1
3175      1    1
3176      1    1
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3183      2    0
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3186      1    1
3187      1    1
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3190      1    1
3191      1    1
3192      1    1
3193      1    1
3194      1    1
3195      2    0
3196      1    1
3197      1    1
3198      1    1
3199      1    1
3200      1    1
3201      3    0
3202      1    1
3203      1    1
3204      1    1
3205      1    1
3206      2    0
3207      1    1
3208      3    0
3209      1    1
3210      1    1
3211      1    1
3212      2    0
3213      2    0
3214      2    0
3215      1    1
3216      1    1
3217      1    1
3218      1    1
3219      1    1
3220      1    1
3221      1    1
3222      2    0
3223      1    1
3224      2    0
3225      1    1
3226      1    1
3227      1    1
3228      2    0
3229      1    1
3230      1    1
3231      1    1
3232      1    1
3233      2    0
3234      1    1
3235      1    1
3236      1    1
3237      2    0
3238      2    0
3239      1    1
3240      1    1
3241      3    0
3242      1    1
3243      2    0
3244      1    1
3245      1    1
3246      1    1
3247      2    0
3248      2    0
3249      1    1
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3251      1    1
3252      1    1
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3258      1    1
3259      1    1
3260      2    0
3261      1    1
3262      1    1
3263      1    1
3264      2    0
3265      1    1
3266      1    1
3267      1    1
3268      1    1
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3270      1    1
3271      1    1
3272      1    1
3273      1    1
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3275      2    0
3276      1    1
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3279      2    0
3280      2    0
3281      1    1
3282      1    1
3283      1    1
3284      1    1
3285      1    1
3286      1    1
3287      1    1
3288      1    1
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3290      1    1
3291      1    1
3292      1    1
3293      2    0
3294      1    1
3295      1    1
3296      1    1
3297      1    1
3298      1    1
3299      1    1
3300      1    1
3301      2    0
3302      2    0
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3307      1    1
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3309      1    1
3310      1    1
3311      2    0
3312      1    1
3313      1    1
3314      1    1
3315      2    0
3316      1    1
3317      1    1
3318      1    1
3319      1    1
3320      1    1
3321      1    1
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3324      1    1
3325      1    1
3326      1    1
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3329      1    1
3330      1    1
3331      1    1
3332      1    1
3333      1    1
3334      2    0
3335      1    1
3336      1    1
3337      1    1
3338      1    1
3339      1    1
3340      1    1
3341      3    0
3342      1    1
3343      1    1
3344      1    1
3345      1    1
3346      1    1
3347      1    1
3348      1    1
3349      1    1
3350      1    1
3351      1    1
3352      1    1
3353      3    0
3354      2    0
3355      1    1
3356      1    1
3357      1    1
3358      1    1
3359      2    0
3360      1    1
3361      2    0
3362      2    0
3363      1    1
3364      1    1
3365      3    0
3366      1    1
3367      1    1
3368      1    1
3369      1    1
3370      1    1
3371      1    1
3372      1    1
3373      2    0
3374      2    0
3375      1    1
3376      2    0
3377      2    0
3378      1    1
3379      1    1
3380      1    1
3381      1    1
3382      2    0
3383      1    1
3384      1    1
3385      1    1
3386      3    0
3387      1    1
3388      1    1
3389      1    1
3390      2    0
3391      1    1
3392      1    1
3393      1    1
3394      2    0
3395      1    1
3396      1    1
3397      1    1
3398      1    1
3399      1    1
3400      1    1
3401      1    1
3402      1    1
3403      1    1
3404      1    1
3405      1    1
3406      1    1
3407      1    1
3408      1    1
3409      1    1
3410      3    0
3411      2    0
3412      1    1
3413      1    1
3414      1    1
3415      1    1
3416      1    1
3417      1    1
3418      1    1
3419      1    1
3420      3    0
3421      1    1
3422      1    1
3423      2    0
3424      1    1
3425      1    1
3426      1    1
3427      1    1
3428      1    1
3429      1    1
3430      1    1
3431      1    1
3432      1    1
3433      2    0
3434      1    1
3435      1    1
3436      1    1
3437      1    1
3438      1    1
3439      2    0
3440      1    1
3441      1    1
3442      1    1
3443      1    1
3444      1    1
3445      1    1
3446      1    1
3447      1    1
3448      1    1
3449      1    1
3450      1    1
3451      1    1
3452      1    1
3453      2    0
3454      1    1
3455      1    1
3456      1    1
3457      1    1
3458      1    1
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3460      2    0
3461      2    0
3462      1    1
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3464      1    1
3465      1    1
3466      1    1
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3469      1    1
3470      1    1
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3472      1    1
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3474      2    0
3475      1    1
3476      1    1
3477      1    1
3478      1    1
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3482      2    0
3483      1    1
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3485      3    0
3486      1    1
3487      1    1
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3490      1    1
3491      1    1
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3493      1    1
3494      1    1
3495      3    0
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3497      1    1
3498      1    1
3499      1    1
3500      1    1
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3502      1    1
3503      1    1
3504      1    1
3505      2    0
3506      3    0
3507      2    0
3508      1    1
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3510      1    1
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3512      1    1
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3516      1    1
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3562      1    1
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3565      1    1
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3641      1    1
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3650      3    0
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3700      3    0
3701      1    1
3702      1    1
3703      1    1
3704      1    1
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3706      1    1
3707      1    1
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3709      1    1
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3716      1    1
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3718      1    1
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3720      1    1
3721      1    1
3722      1    1
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3725      1    1
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3730      3    0
3731      1    1
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3733      1    1
3734      1    1
3735      1    1
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3738      1    1
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3747      1    1
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3750      1    1
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3755      1    1
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3761      1    1
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3767      1    1
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3773      1    1
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3780      1    1
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3782      1    1
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3784      1    1
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3786      1    1
3787      1    1
3788      1    1
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3790      1    1
3791      3    0
3792      1    1
3793      2    0
3794      1    1
3795      1    1
3796      1    1
3797      1    1
3798      1    1
3799      1    1
3800      1    1
3801      1    1
3802      2    0
3803      3    0
3804      1    1
3805      2    0
3806      1    1
3807      1    1
3808      1    1
3809      1    1
3810      1    1
3811      1    1
3812      1    1
3813      1    1
3814      1    1
3815      1    1
3816      1    1
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3819      1    1
3820      1    1
3821      1    1
3822      1    1
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3824      3    0
3825      1    1
3826      1    1
3827      1    1
3828      3    0
3829      1    1
3830      1    1
3831      1    1
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3833      1    1
3834      3    0
3835      3    0
3836      2    0
3837      2    0
3838      1    1
3839      2    0
3840      2    0
3841      1    1
3842      1    1
3843      1    1
3844      2    0
3845      1    1
3846      1    1
3847      1    1
3848      1    1
3849      1    1
3850      1    1
3851      1    1
3852      1    1
3853      2    0
3854      1    1
3855      3    0
3856      2    0
3857      2    0
3858      3    0
3859      1    1
3860      1    1
3861      1    1
3862      2    0
3863      1    1
3864      1    1
3865      1    1
3866      1    1
3867      2    0
3868      1    1
3869      1    1
3870      1    1
3871      1    1
3872      1    1
3873      1    1
3874      1    1
3875      1    1
3876      1    1

Which of the following implies i.i.d?

Option 1

x = sample(gss$vote, size = 10, replace = FALSE)

Option 2

x = sample(gss$vote, size = 10, replace = TRUE)

Option 3

x0 = sample(gss$vote, size = 1)
x1 = sample(gss$vote, size = 1)
x2 = sample(gss$vote, size = 1)
x3 = sample(gss$vote, size = 1)
x4 = sample(gss$vote, size = 1)
x5 = sample(gss$vote, size = 1)
x6 = sample(gss$vote, size = 1)
x7 = sample(gss$vote, size = 1)
x8 = sample(gss$vote, size = 1)
x9 = sample(gss$vote, size = 1)

Pretend our data is the population

Population mean

mean(gss$vote)

[1] 0.7425181

Sample mean

set.seed(20241009)
x = sample(gss$vote, size = 10, replace = FALSE)
mean(x)

[1] 0.5

Bias

mean(x) - mean(gss$vote)

[1] -0.2425181

MSE

mean((x - mean(gss$vote))^2)

[1] 0.308815

Finite-sample properties

• Code
• Output
• Figure

sample_mean = function(n){
  mean(sample(gss$vote, size = n), replace = FALSE)
}

set.seed(20241009)
means = replicate(10, sample_mean(100))

means

 [1] 0.72 0.79 0.70 0.72 0.71 0.77 0.71 0.73 0.72 0.78

ggplot(mapping = aes(x = means)) +
  geom_density() +
  geom_vline(xintercept = mean(gss$vote), linetype = "dashed") +
  xlim(0,1)

Increase the number of replicates

• 10 replications
• 100 replications
• 1000 replications

Asymptotic properties

• Function
• Scale up
• Plot

draw_sample = function(data, n) {
  df = data %>% 
    sample_n(size = n) %>% 
    summarize(
      estimate = mean(vote),
      std.error = sd(vote),
      rmse = sqrt(mean((vote - mean(gss$vote))^2))
    )
  
  return(df)
}

# Example
draw_sample(gss, 100)

# A tibble: 1 × 3
  estimate std.error  rmse
     <dbl>     <dbl> <dbl>
1     0.74     0.441 0.439

n = seq(10, 1000, by = 10)

set.seed(20241009)
reps = n %>% 
  map(~draw_sample(gss, n = .)) %>% 
  bind_rows %>% 
  mutate(n = n)

head(reps)

# A tibble: 6 × 4
  estimate std.error  rmse     n
     <dbl>     <dbl> <dbl> <dbl>
1    0.5       0.527 0.556    10
2    0.8       0.410 0.404    20
3    0.633     0.490 0.494    30
4    0.8       0.405 0.404    40
5    0.72      0.454 0.450    50
6    0.867     0.343 0.362    60

Variance

Why is there a difference?

set.seed(20241009)
gss %>%
  # replace = FALSE by default
  sample_n(size = 400) %>% 
  summarize(var1 = var(vote),
            var2 = mean(mean(vote^2) - mean(vote)^2),
            var3 = mean(mean(vote^2) - mean(vote)^2) * (400/(400-1))
            )

# A tibble: 1 × 3
   var1  var2  var3
  <dbl> <dbl> <dbl>
1 0.198 0.197 0.198

Properties of estimators

• Unbiasedness: \(E[\widehat \theta] = \theta\)

• Efficiency: \(\widehat \theta_A\) is more efficient that \(\widehat \theta_ B\) if it has lower MSE

• Consistency: \(\widehat \theta \overset{p}{\to} \theta\)

Next week: Regression (AM Chapter 4)

Important:

• Estimation (Connection to BLP)
• Inference (3 types of standard errors)
• Application (Clean water and infant mortality)

Less important (POLI_SCI 405):

• Polynomials
• Interactions
• Overfitting
• Average partial derivatives vs. partial derivative at the average (aka marginal effects)

Ignore:

• Sieve estimation
• Penalized regression

Next week: Other readings

Berk (2010)

• Regression level I, II, III

Achen (2005)

• Why are “garbage-can” regressions a problem?
• Total vs direct effects
• Linearity vs. monotonicity

Hansen (2022) (skip proofs)

• What does it mean for OLS to be BUE instead of BLUE? Why does it matter?
• History of OLS