Base Shifting

We can do base shifting using the go-to splicing method:

Tldr

Just remember one basic thing: The base might change the absolute value, but the factor of growth between subsequent years will remain the same.

Consider the Consumer Price Index data (from RBI). They provided parts of the data to use in different base years.

Data (click to expand)

Consumer Price Index

Year	(1960 = 100)	(1982=100)	(2001=100)	(2016=100)
1970-71	186
1971-72	192
1972-73	207
1973-74	250
1974-75	317
1975-76	313
1976-77	301
1977-78	324
1978-79	331
1979-80	360
1980-81	401
1981-82	451
1982-83	486
1983-84	547	111
1984-85	582	118
1985-86	620	126
1986-87	674	137
1987-88	736	149
1988-89		163
1989-90		173
1990-91		193
1991-92		219
1992-93		240
1993-94		258
1994-95		284
1995-96		313
1996-97		342
1997-98		366
1998-99		414
1999-00		428
2000-01		444
2001-02		463	100
2002-03		482
2003-04		500
2004-05		520
2005-06			117
2006-07			125
2007-08			133
2008-09			145
2009-10			163
2010-11			180
2011-12			195
2012-13			215
2013-14			236
2014-15			251
2015-16			265
2016-17			276	100
2017-18			284
2018-19			300
2019-20			323
2020-21				118
2021-22				124
2022-23				131
2023-24				138

But, recall the core fundamental, the factor of growth between any two subsequent years will remain the same.

So now, consider the year range ($1983\to 1988$), and calculate the factors of growth between subsequent years, in both the bases.

Year	$X_t$ (1960=100)	$Y_t$ (1982=100)	$X_{t+1}/X_t$	$Y_{t+1}/Y_t$
1983-84	547	111	1.063063	1.063985
1984-85	582	118	1.067797	1.065292
1985-86	620	126	1.087302	1.087097
1986-87	674	137	1.087591	1.091988
1987-88	736	149

We note that the factors are almost same, but understand that they are in reality, exactly the same and due to the fact that CPIs are rounded to the nearest units digit, we get that difference.

So, what do we need to do in order to extrapolate values?

Just multiply the given year $Z_t$ (in any base) with the factor of growth for that particular year $\frac{X_{t+1}}{X_t}$ (for any other base). For example, to calculate $Z_{2016}$, if $Z_{2015}$ is given we just need to do

$$Z_{2016}=Z_{2015}\times \frac{X_{2016}}{X_{2015}}$$

Please understand that this works, because the factor of growth from 2015 to 2016 is the same whether we calculate it wrt. the base year for the $X$-series or the $Z$-series. It’s all the same.