My MacBook air is frozen while installing the new update...

Possible Fixes for El Capitan and Later Installations

You should try each, one at a time, then test to see if the problem is fixed before going on to the next.

Be sure to backup your files before proceeding.

1. Resetting your Mac’s PRAM and NVRAM
2. Reset the System Management Controller (SMC)
3. Start the computer in Safe Mode, then restart normally. This is slower than a standard startup.
4. Repair the disk by booting the from the Recovery HD. Immediately after the chime hold down the Command and R keys until the Utility Menu appears. Choose Disk Utility and click on the Continue button. Select the indented (usually, Macintosh HD) volumeentry from the side list. Click on the First Aid button in the toolbar. Wait for the Done button to appear. Quit Disk Utility and returnto the Utility Menu. Restart the computer from the Apple Menu.
5. Install Combo Updater Download macOS Sierra 10.12.2 Combo Update and install.
6. Reinstall OS X by booting from the Recovery HD using the Command and R keys. When the Utility Menu appears select Reinstall OS X then click on the Continue button.
7. Erase and Install OS X Restart the computer. Immediately after the chime hold down the Command and R keys until the Apple logo appears. When the Utility Menu appears:

1. Select Disk Utility from the Utility Menu and click on Continue button.
2. When Disk Utility loads select the drive (out-dented entry) from the Device list.
3. Click on the Erase icon in Disk Utility's toolbar. A panel will drop down.
4. Set the Format type to Mac OS Extended (Journaled.)
5. Click on the Apply button, then wait for the Done button to activate and click on it.
6. Quit Disk Utility and return to the Utility Menu.
7. Select Reinstall OS X and click on the Continuebutton.

Sometimes a shutdown using the power switch and then a restart will let the installation continue. What update?

When ever an update is there in App Store , there is an option written as More........ just click on More & it will prompt to download & Restart .So click on restart let this be updated , after that go for second update and follow the same procedure .

Or sometimes only updates is shown in that case click on update , it will not prompt to restart .

Restarting is must , I am doing it and this method always worked for me , some users use combo updates .

Network speed has a big role , the faster the speed the better are the results in updation .