Germán Rodríguez
Demographic Methods Princeton University

## Population Projections

This computing log illustrates the use of the cohort component method in population projections.

We will use the data from Sweden used in Box 6.1 (page 125 in the textbook), available as file `sweden93.dat` in the datasets section to save you some typing. We read the population counts, the life table, and the fertility rates. The last age group is 85+ and `L` for that age is actually time lived after age 85. We also express person-years per birth, dividing by l0.

```. infile age p L f using ///
>    https://data.princeton.edu/eco572/datasets/sweden93.dat

. mata
------------------------------------------------- mata (type end to exit) --------------
:         p93 = st_data(., "p")

:         L = st_data(., "L")/100000

:         f = st_data(., "f")

:         sum(p93)
4397428

: end
----------------------------------------------------------------------------------------
```
```> sw <- read.table("https://data.princeton.edu/eco572/datasets/sweden93.dat", header=FALSE)

> names(sw) <- c("age", "p93", "L", "f")

> sw <- mutate(sw, L = L/100000)

> summarize(sw, sum(p93))
sum(p93)
1  4397428
```

### The Leslie Matrix

We will write a function to compute a Leslie matrix given two vectors representing person-years lived 5Lx (with the convention that the radix is one and the last value is Tx) an the maternity rates 5mx. (To simplify cutting and pasting I provide the code rather than a log.)

```    real matrix Leslie(real vector L, real vector m) {
n = length(L)
M = J(n,n,0)

// lower diagonal has survivorship ratios
for (i=1; i < n; i++) {
M[i+1,i] = L[i+1]/L[i]
}
M[n,n-1] = M[n,n] = L[n]/(L[n-1]+L[n])

// first row has net maternity contributions
for(i=1; i < n; i++) {
if(m[i]==0 & m[i+1]==0) continue
M[1,i] = L*(m[i] + m[i+1]*L[i+1]/L[i])/2
}
if (m[n] > 0) M[1,n] = L*m[n]
return(M)
}
```
```    Leslie <- function(L, m) {
n = length(L)
M = matrix(0, n, n)

# lower diagonal has survivorship ratios
for (i in 1:(n-1)) {
M[i+1,i] <- L[i+1]/L[i]
}
M[n,n-1] <- M[n,n] <- L[n]/(L[n-1] + L[n])

# first row has net maternity contributions
for(i in 1:(n-1)) {
if(m[i] != 0 | m[i+1] != 0) {
M[1,i] <- L * (m[i] + m[i+1] * L[i+1]/L[i])/2
}
}
if (m[n] > 0) M[1,n] <- L * m[n]
M
}
```

The function computes the survivorship ratios and stores them in the lower diagonal. For example `M[2,1] = L/L` is the probability of surviving from age 0-4 to 5-9, or 5L5/5L0 in standard demographic notation.

The only tricky bit is the last (open) age interval, where we combine the last two groups and project then using `L[n]/(L[n-1] + L[n])`. In Sweden we combine ages 80-84 and 85+ and use T85/T80 = T85/(5L80+T80) as the survival ratio. (As noted in class, if we had T90 we would use a slightly different procedure. The textbook describes that procedure on page 121, but uses the combined projection in Box 6.1.)

The rest of the calculation computes the average fertility rate for each age group and then survives the resulting births to age 0-4. For example women 15-19 are exposed to the rates at 15-19 and 20-24, with the latter discounted by the probability of surviving to 20-24, so we average `m` and `m*L/L`.

The rate would be multiplied by 5, the width of the period, and by the probability of surviving from birth to age 0-4, which is `L/5`. The 5's cancel, so we don't include them. (As noted in class there are two ways of computing births; here we focus on the women and average the rates, an approach best suited for computing the Leslie matrix. The textbook also describes focusing on the rates and averaging the numbers of women exposed to each rate.)

### Projections for 5 and 10 years

For Sweden we have fertility rates, so we divide by 2.05 to obtain maternity rates (female births) before calling our function. Once we have a Leslie matrix, projection is easy

```. mata
------------------------------------------------- mata (type end to exit) --------------
:         M = Leslie(L, f/2.05)

:         p98 = M * p93

:         p03 = M * p98

:         round( (p93, p98, p03) )
1        2        3
+----------------------------+
1 |  293395   293574   280121  |
2 |  248369   293189   293368  |
3 |  240012   248251   293049  |
4 |  261346   239833   248066  |
5 |  285209   261015   239529  |
6 |  314388   284787   260629  |
7 |  281290   313782   284238  |
8 |  286923   280463   312859  |
9 |  304108   285576   279147  |
10 |  324946   301731   283344  |
11 |  247613   320974   298043  |
12 |  211351   243039   315045  |
13 |  215140   205109   235861  |
14 |  221764   204944   195388  |
15 |  223506   204793   189260  |
16 |  183654   194419   178141  |
17 |  141990   142324   150666  |
18 |  112424   131768   141960  |
+----------------------------+

:         sum(p98), sum(p03)
1            2
+---------------------------+
1 |  4449569.73   4478712.28  |
+---------------------------+

: end
----------------------------------------------------------------------------------------
```
```> M <- Leslie(sw\$L, sw\$f/2.05)

> sw <- mutate(sw,
+   p98 = M %*% p93,
+       p03 = M %*% p98)

> round(select(sw, age, p93, p98, p03), 0)
age    p93    p98    p03
1    0 293395 293574 280121
2    5 248369 293189 293368
3   10 240012 248251 293049
4   15 261346 239833 248066
5   20 285209 261015 239529
6   25 314388 284787 260629
7   30 281290 313782 284238
8   35 286923 280463 312859
9   40 304108 285576 279147
10  45 324946 301731 283344
11  50 247613 320974 298043
12  55 211351 243039 315045
13  60 215140 205109 235861
14  65 221764 204944 195388
15  70 223506 204793 189260
16  75 183654 194419 178141
17  80 141990 142324 150666
18  85 112424 131768 141960

> select(sw, p93, p98, p03) %>% colSums()
p93     p98     p03
4397428 4449570 4478712
```

These results agree exactly with Box 6.1 (part 2) in the text.

### The Stable Equivalent

While we have the Leslie matrix handy we can compute the intrinsic rate of growth and the stable age distribution by simply obtaining the first eigenvalue and eigenvector of the projection matrix.

This is easy to do in Mata, as long as we define the output vector and matrix before. Here I use empty matrices.

```. mata
------------------------------------------------- mata (type end to exit) --------------
:         values = J(0,0,.)

:         vectors = J(0,0,.)

:         eigensystem(M,vectors,values)

:         values
1.00111254

:         log(values)/5
.000222384

:         stable = Re(vectors[,1]/sum(vectors[,1]))

:         stable
1
+---------------+
1 |  .0619116758  |
2 |   .061799489  |
3 |  .0617013824  |
4 |  .0615869209  |
5 |  .0614405485  |
6 |  .0612814306  |
7 |  .0610952702  |
8 |  .0608479528  |
9 |  .0604950315  |
10 |  .0599554176  |
11 |  .0591567526  |
12 |  .0579994633  |
13 |  .0562239225  |
14 |  .0534997793  |
15 |  .0493506336  |
16 |   .042880453  |
17 |  .0331935312  |
18 |  .0355803452  |
+---------------+

: end
----------------------------------------------------------------------------------------
```
```> e <- eigen(M)

> e\$values
 1.001113+0i

> log(e\$values)/5
 0.0002223831+0i

> arv <- abs(Re(e\$vectors[,1]))

> stable <- arv/sum(arv)

> stable
 0.06191167 0.06179949 0.06170138 0.06158692 0.06144055 0.06128143
 0.06109527 0.06084795 0.06049503 0.05995542 0.05915675 0.05799946
 0.05622392 0.05349978 0.04935063 0.04288045 0.03319353 0.03558035
```

The first eigenvalue is 1.0011, so if the 1993 fertility and mortality rates were to stay constant the population would eventually grow 0.11 percent every 5 years, or 0.022 percent per year. The first eigenvector, divided by its sum, gives us the proportionate age distribution. In general the eigenvalues and eigen vectors can be complex numbers and both Stata and R store them as such, although for a Leslie matrix they are always real. The function `Re()` takes the real part of a complex number.

We return to stable populations in a separate handout, but there is a lot you could do with the Swedish data. For one thing you might try projecting the population for 100 years of so (20 periods) to verify that it becomes stable. You may also try plotting the current and stable equivalent age distributions to see what that says about population momentum.