Time series data vs. panel data

Question

What is the difference between time series data and panel data? I think that they are both multi-dimensional data, and both of them involve measurements over time. Is it that the covariance is constant in time series data, but not in panel data?

Answer 1

A typical linear model for time series might look like this:

$$ y_t = \beta_0 + \beta_1 x_t + u_t $$

That is, a model is indexed over time $t$. You have a unit of observation, and you follow this unit over time. For panel data you follow multiple units over time, and a model might be:

$$ y_{it} = \beta_0 + \beta_1 x_{it} + u_{it} $$

Notice how now we have both a unit index $i$ and a time index $t$. Notice that this differences comes regardless of any assumptions we might impose on the data.

You mention specially "that the covariance is constant in time series data", in general there is no reason to assume this (and this assumption does make time series... a time series). But it is fair to say that most time series methods impose this assumption in some way.

Another way to see this is look at some raw data. Here is some time series data (from R):

# Make data:
set.seed(42)
time.series <- data.frame(year = 1991:2000, gdp = abs(rnorm(10,50,5)), capita = 50000 + rnorm(10))
time.series

   year      gdp   capita
1  1991 56.85479 50001.30
2  1992 47.17651 50002.29
3  1993 51.81564 49998.61
4  1994 53.16431 49999.72
5  1995 52.02134 49999.87
6  1996 49.46938 50000.64
7  1997 57.55761 49999.72
8  1998 49.52670 49997.34
9  1999 60.09212 49997.56
10 2000 49.68643 50001.32

And some panel data might look like this:

require(plm)
data("Produc", package = "plm")
head(Produc, n = 20)

      state year region     pcap     hwy   water    util       pc   gsp    emp unemp
1  ALABAMA 1970      6 15032.67 7325.80 1655.68 6051.20 35793.80 28418 1010.5   4.7
2  ALABAMA 1971      6 15501.94 7525.94 1721.02 6254.98 37299.91 29375 1021.9   5.2
3  ALABAMA 1972      6 15972.41 7765.42 1764.75 6442.23 38670.30 31303 1072.3   4.7
4  ALABAMA 1973      6 16406.26 7907.66 1742.41 6756.19 40084.01 33430 1135.5   3.9
5  ALABAMA 1974      6 16762.67 8025.52 1734.85 7002.29 42057.31 33749 1169.8   5.5
6  ALABAMA 1975      6 17316.26 8158.23 1752.27 7405.76 43971.71 33604 1155.4   7.7
7  ALABAMA 1976      6 17732.86 8228.19 1799.74 7704.93 50221.57 35764 1207.0   6.8
8  ALABAMA 1977      6 18111.93 8365.67 1845.11 7901.15 51084.99 37463  1269.2   7.4
9  ALABAMA 1978      6 18479.74 8510.64 1960.51 8008.59 52604.05 39964 1336.5   6.3
10 ALABAMA 1979      6 18881.49 8640.61 2081.91 8158.97 54525.86 40979 1362.0   7.1
11 ALABAMA 1980      6 19012.34 8663.50 2138.52 8210.33 56589.16 40380 1356.1   8.8
12 ALABAMA 1981      6 19118.52 8628.83 2218.91 8270.79 56481.93 41105 1347.6  11.0
13 ALABAMA 1982      6 19118.25 8645.14 2215.84 8257.26 58021.69 40328 1312.5  14.0
14 ALABAMA 1983      6 19122.00 8612.47 2230.91 8278.63 58893.97 42245 1328.8  14.0
15 ALABAMA 1984      6 19257.47 8655.94 2235.16 8366.37 59446.86 45118 1387.7  11.0
16 ALABAMA 1985      6 19433.36 8726.24 2253.03 8454.09 60688.04 46849 1427.1   8.9
17 ALABAMA 1986      6 19723.37 8813.24 2308.99 8601.14 61628.88 48409 1463.3   9.8
18 ARIZONA 1970      8 10148.42 4556.81 1627.87 3963.75 23585.99 19288  547.4   4.4
19 ARIZONA 1971      8 10560.54 4701.97 1627.34 4231.23 24924.82 21040  581.4   4.7
20 ARIZONA 1972      8 10977.53 4847.84 1614.58 4515.11 26058.65 23289  646.3   4.2

See the way the data is stacked? Here "state" identifies $i$ from the above equation, and "year" identifies $t$.